Which realm has the best archers in Westeros?

Well the other answer may have addressed a possible house, but I'm just going to reveal the most canonical answer we have.

_{Full credits to Aegon for the image}

No the answer is NOT the Greyjoys! We simply do not semear conhecer

Largest regiments of archers

• Raven's Teeth

  The Raven's teeth are the largest regiment of archers we come to meet. Being loyal to the crown and Bloodraven's personal escort, they were based in King's Landing in the Crownlands. The extent to which their makeup was from the Crownlands is unknown, but we can guess that many would be from lesser houses in the Crownlands, and not necessarily from other realms (as those Lord would likely keep good archers to themselves). This force was 500 men strong during the First Blackfyre Rebellion at the Redgrass field. At the Second Blackfyre rebellion, Bloodraven only took part of his escort to Whitewalls, which consisted of 300 Raven's Teeth. And finally as noted below, a portion of the Raven's Teeth went with Brynden Rivers when he was sent to the wall, this group included 200 men.

• The Wall

  As we see from some of the men listed below, the wall had many bowmen and some of particularly great renown (at least amongst the Black Brothers). We know that bowmen are useful from shooting down off the top of the wall.

  They'd hung a target on the granary doors, so they could seem to be honing their skills as archers, but he knew lurkers when he saw them.

  Most of these archers were most likely trained at the Wall, as in later years most recruits were:

  a pack of thieves, killers, and baseborn churls.
  _{A Storm of Swords - Tyrion IV}

• The Lannisters

  They're said to have foot archers (3 lines worth) led by Kevan Lannister. What types of bows they used is unknown.

  His uncle would lead the center. Ser Kevan had raised his standards above the kingsroad. Quivers hanging from their belts, the foot archers arrayed themselves into three long lines, to east and west of the road, and stood calmly stringing their bows.

• The Golden Company

  I won't go into extensive detail here as I don't want to be accused for stealing an answer that was posted below. (Although the relevant quotes can be found under "Black Balaq") The Golden company contained 1000 archers led by Black Balaq, a third of which were Westerosi born. Although they are currently in the Stormlands and as of Os Ventos do inverno have taken Storm's End, they were trained in the Free Cities and not in Westeros and therefore can't be counted amongst the best "trained" archers of a realm in Westeros.

Quality of equipment

The longest range bows in Planetos are Dragonbone, "greatly prized by the Dothraki" followed by those of the Goldenheart tree, used by the Summer Islanders.

However we care about westeros, where the only bows that seems to get mentioned are Dornish yew Longbows, used by the Westerosi in the Golden Company, and those made of Weirwood, as used by Bryden Rivers and Ygritte. Dorne and the North therefore seem to be the most equipt for making the best bows.

• Anguy (Dornish Marches/Stormlands)

  That afternoon a boy named Anguy, an unheralded commoner from the Dornish Marches, won the archery competition, outshooting Ser Balon Swann and Jalabhar Xho at a hundred paces after all the other bowmen had been eliminated at the shorter distances. Ned sent Alyn to seek him out and offer him a position with the Hand's guard, but the boy was flush with wine and victory and riches undreamed of, and he refused.
  _{A Game of Thrones - Eddard VII}

  As they waded across, their singing flushed a duck from among the reeds. Anguy stopped where he stood, unslung his bow, notched an arrow, and brought it down. The bird fell in the shallows not far from the bank.
  _{A Storm of Swords - Arya II}

• Balon Swann (Dornish Marches/Stormlands)

  As is said above, he came second in the Hand's Tourney, furthermore we have the following from Um choque de reis

  In addition to being well born, young Ser Balon was valiant, courtly, and skilled at arms; good with a lance, better with a morningstar, superb with the bow.
  _{A Clash of Kings - Tyrion XI}

• Squire Dalbridge (Crownlands, 0.5)

  Due to the possibility of Squire Dalbridge being from the Crownlands, as he was squire to King Jaehaerys II Targaryen, I'm going to count him as half a point.

  Said to have the keenest eyes in the watch. The squire was also good with a bow.

  Should I need to put an arrow through the eye of some foe across a windy battlefield, I summon Squire Dalbridge.
  _{A Clash of Kings - Jon VII}

• Alleras (Dorne/Summer Islands)

  "There's another apple near your foot," Alleras called to Mollander, "and I still have two arrows in my quiver." "Fuck your quiver." Mollander scooped up the windfall. "This one's wormy," he complained, but he threw it anyway. The arrow caught the apple as it began to fall and sliced it clean in two.
  _{A Feast for Crows - Prologue}

• Fletcher Dick (Dornish Marches/The Stormlands)

  Fletcher Dick it was who showed me how to bend the bow, and no finer archer ever lived.
  _{A Storm of Swords - Samwell II}

  Fletcher Dick, the notorious outlaw of the Kingswood Brotherhood, was born in a village near the Marcher castle of Stonehelm, and is held by many to be the finest archer who ever drew bow.
  _{The World of Ice and Fire - The Stormlands: The Men of the Stormlands}

• Ulmer (Crownlands/Stormlands), Donnel Hill (Westerlands, 0.5), Muttering Bill (Unknown), Garth Greyfeather (Unknown) , Ronnel Harclay (The North)

  Muttering Bill and Garth Greyfeather have all been left as 'unknown'. For the latter two, we know nothing about the past. Ulmer however, has been marked as from the Crownlands/Stormlands since he was part of the Kingswood Brotherhood and learnt to use a bow from Fletcher Dick. Donnel Hill has received 0.5 points for the Westerlands as he claims to be a bastard of House Lannister, however whether he truly is from there or how much archery he learnt there and how much he learnt at the wall. Ronnel Harclay is of House Harclay in the North.

  while Mormont had divided the arrows among his best bowmen. Muttering Bill, Garth Greyfeather, Ronnel Harclay, Sweet Donnel Hill, and Alan of Rosby had three apiece, and Ulmer had four.
  _{A Storm of Swords - Samwell II}

• Brynden Tully (Riverlands)

  Swiftly,” Ser Brynden said. He nocked an arrow, held it steady for the brand, drew and released before Catelyn was quite sure that the fire had caught . . . but as the shot rose, she saw the flames trailing through the air, a pale orange pennon. The boat had vanished in the mists. Falling, the flaming arrow was swallowed up as well . . . but only for a heartbeat. Then, sudden as hope, they saw the red bloom flower.
  _{A Storm of Swords - Cately IV}

• Bloodraven, Bryden Rivers (Crownlands)

  Brynden Rivers was the Hand of the King under Aerys I Targaryen and Maekar I Targaryen and later also the Lord Commander of the Night's Watch and the Three-eyed Crow

  Although I struggled to find explicit evidence that Bloodraven is skilled with a bow, we have the following about the end of the First Blackfyre Rebellion and that he commanded an army of archers known as the "Raven's Teeth".

  ...but it was the kinslayer who turned the tide with a white arrow and a black spell.
  _{The Dunk & Egg Novellas - The Sworn Sword}

  Daemon and his eldest sons, Aegon and Aemon, were brought down beneath the withering fall of arrows sent by Brynden Rivers and his private guards, the Raven's Teeth.
  _{The World of Ice and Fire - The Targaryen Kings: Daeron II}

  Many of which he took with him to The Wall in 233AC, at the same time Maester Aemon travelled north.

  Ser Brynden Rivers set sail for the Wall late in the year of 233 AC. (No one intercepted his ship). Two hundred men went with him, many of them archers from Bloodraven's personal guard, the Raven's Teeth.
  _{The World of Ice and Fire - The Targaryen Kings: Aegon V}

• Theon Greyjoy (Iron Islands/The North)

  I've counted Theon as both as although he was born in the Iron Islands, he was raised as a ward of Eddard Stark in Winterfell and likely gained prowess with a bow there. The Iron Islanders are not known for having any particular skills with a bow.

• Foss the Archer (The Reach)

  Foss the Archer, renowned for shooting apples off the head of any maid who took his fancy, from whom both the red apple and green apple Fossoways trace their descent.
  _{The World of Ice and Fire - The Reach: Garth Greenhand}

Below are noted the Summer Islanders, although not of Westeros they are believed to have done trade with Westeros from before the First Men came across the Narrow Sea.

• Kojja Mo (Summer Islands)

  Kojja Mo, Daughter of Quhuru Mo, Captain of the Cinnamon Wind, was known for her and her archer's prowess with a bow. The used "great bows of goldenheart that could send a shaft further and truer than even Dornish yew"

  She captained the ship's red archers too, and pulled a double-curved goldenheart bow that could send a shaft four hundred yards.
  _{A Feast for Crows - Samwell IV}

  The only thing Kojja Mo loved better than her bow was bouncing Dalla's boy upon her knee and singing to him in the Summer Tongue.
  _ibid.

• Jalabhar Xho (Summer Islands)

  Coming second to Anguy in the Archery competition and the Hand's Tourney, Jalabhar Xho is one of the infamous archers from the Summer Islands and an exiled Prince of the Red Flower Vale

• Black Balaq (Summer Islands)

  A captain of the Golden Company, Black Balaq led a thousand seasoned archers of various origin.

  Black Balaq commanded one thousand bows.

  A third of Balaq's men used crossbows, another third the double-curved horn-and-sinew bows of the east. Better than these were the big yew longbows borne by the archers of Westerosi blood, and best of all were the great bows of goldenheart treasured by Black Balaq himself and his fifty Summer Islanders. Only a dragonbone bow could outrange one made of goldenheart. Whatever bow they carried, all of Balaq's men were sharp-eyed, seasoned veterans who had proved their worth in a hundred battles, raids, and skirmishes.
  _{A Dance with Dragons - The Griffin Reborn}

  Black Balaq also showed his prowess with a bow, feathering a Raven as it left the Maester's tower at Griffin's Roost.

  As they neared the castle, he saw a third raven flap from the maester's tower, only to be feathered by Black Balaq himself.
  _ibid.

The current standings are:

• Westeros
  • Stormlands (4)
  • Dornish Marches (3)
  • Crownlands (2.5)
  • The North (2)
  • The Iron Islands (1)
  • Riverlands (1)
  • The Reach (1)
  • Dorne (1)
  • Westerlands (0.5)
  • Unknown (4)
• Planetos
  • Summer Islands (4)

From the above tables. We can see that the Stormlands has produced the most talent, of which the Dornish Marches (part of the Stormlands and the Reach) were the key provider of skill. Dorne proper, however only produced one renowned archer, as did various other realms. Considering we have 4 unknowns, any of the realms could take the top spot and be the greatest. So I'm going to conclude with a bit of conjecture, but considering the Summer Islanders did trade with Westeros from before the first men, and Landed at Old Town and King's Landing, it is likely their trade was taught around nearby southern ports. Leading on to my additional bit.

Finally, the Summer Islands are infamous for their archery, although they are not part of Westeros and the Seven Kingdoms, they are known for their great skill with a goldenheart bow and producing bows which are second only in range to those made of dragonbone.

List of things I plan to work on: - More on Lannister forces, the Night's Watch and the Bog Devils. - Possibly revised list of Archers and a better explanation of what it means. - Evaluating the history of Archery (if any) in Westeros. - More on the Summer Islanders, Ghiscari, Dothraki and other Essosi and Sothyrosi archers.

I don't know about a whole Realm, but House Sarsfield (Bannermen to House Lannister) seems to have prided itself on its archery.

Their sigil is a green arrow on a white bend on green. And their motto (semi-canon) is "True to the Mark"

I think the best answer is we do not know, just like Edlothiad suggested, however, we need to give special consideration to the Iron Islands.

Aqui, Theon boasts for the archery skills of either Greyjoys or of all Iron Islanders. Moreover, Maester Luwin does not correct him.

This further makes sense, because naval warfare requires extended use of archery compared to land warfare. While Iron Islanders are good with archery, navigation and personal combat, they are not good at horsetop and they are (probably) not good in fighting in formation. Note that all good infantry must fight well in formation, that is the real strength of Unsullied.

The above analyses show that there is no a major archery power within any formerly acknowledged Westerosi House or Reino.

However, as of the end of A Dance with Dragons, the Golden Company landed in Westeros at Griffins Roost in the Stormlands with up to Arqueiros 1000.

1,000 archers, a third of these archers are listed as crossbowmen while another third are said to be using double-curved horn-and-sinew bows of the east. The final part of the archers consist of men with Westerosi blood who use big yew long bows, and fifty Summer Islanders using great bows of Goldenheart.
_{http://awoiaf.westeros.org/index.php/Golden_Company}

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Whatever bow they carried, all of Balaq's (the Golden Company Archery Commander) men were sharp-eyed, seasoned veterans who had proved their worth in a hundred battles, raids, and skirmishes.
_{A Dance with Dragons, Chapter 61, The Griffin Reborn.}

This means they have roughly 300 Westeros Longbowmen which we can assume are very similar to the English Longbowmen. This is by far the biggest contingent of Longbow type archers we hear of and are all of Westerosi blood.

Therefore the best archery force can be found in the Stormlands but we might want to rename this "The Kingdom of Prince Aegon Targaryen".

Bonus Points from the show: We see that the Dothraki that invade with Daenerys have a mounted archery contingent. If there were 100,000 Dothraki screamers with 5% of them owning a bow then we would have a mounted archery division of 5,000 units. This would then be the most significant archery division in Westeros by a significant margin.

Bonus Points from the Books: The Night's Watch has several hundred recruits who must all practicing archery every day. This is reminiscent of the archaic English laws saying boys and young men must practice archery everyday. If we also add in the Wildlings at the wall - many of whom hunt with bow - we might have a >1000 strong archery force in the North.

As well as all the excellent information in @Edlothiad's answer we also learn of at least one other house that is apparently renowned for their archers.

Casa Tarth

"Tyrell swords will make me king. Rowan and Tarly and Caron will make me king, with axe and mace and warhammer. Tarth arrows and Penrose lances, Fossoway, Cuy, Mullendore, Estermont, Selmy, Hightower, Oakheart, Crane, Caswell, Blackbar, Morrigen, Beesbury, Shermer, Dunn, Footly . . . even House Florent, your own wife's brothers and uncles, they will make me king.
_{A Clash of Kings, Catelyn III}

Calling them out specifically for their archery when other houses like Penrose are called out for their lances and Tyrell for their swords makes it seem like they must at least have some good archers.