After both his father and brother (who was the heir) executed in Season 7 of Game of Thrones, how come that Samwell Tarly didn't become the new head of the House, i.e. Lord of Horn Hill?
I would expect the remaining house leaders to have sent Sam a message letting him know of what happened and asking him to come back home to rule the house.
Why didn't it happen, and he wasn't even notified?
Sam is a brother of the Night's Watch and has given up his right of succession as part of his oath.
Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands, father no children. I shall wear no crowns and win no glory. I shall live and die at my post. I am the sword in the darkness. I am the watcher on the walls. I am the shield that guards the realms of men. I pledge my life and honor to the Night's Watch, for this night and all the nights to come.
As for why he wasn't told of their death's well the maester's at the Citadel chose not to tell him "just yet".
MAESTER 1: Is he the one whose father and brother were just burned alive?
MARWYN: I'm afraid so.
MAESTER 2: Horrible business.
MARWYN: I don't have the heart to tell him yet. He's a good lad.
Game of Thrones, Season 7 Episode 5, "Eastwatch"
Sam is in the Night's Watch.
Members renounce all titles and claims to lands, so he can't become a Lord.
Because he has forsaken all claims to that title and any other title the moment he joined the Night's Watch.
Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands, father no children. I shall wear no crowns and win no glory. I shall live and die at my post. I am the sword in the darkness. I am the watcher on the walls. I am the fire that burns against the cold, the light that brings the dawn, the horn that wakes the sleepers, the shield that guards the realms of men. I pledge my life and honor to the Night's Watch, for this night and all the nights to come. - Night's Watch Vows
Not to mention that those who join the Citadel to become Maesters also forsake claims to titles.
When an acolyte of noble birth takes his vows and dons his chain, he puts aside his House name. He swears sacred vows, promising to hold no lands or lordships, and to be celibate. - Maester Vows
He is in the Nights Watch. Remember? He was sent to the Citadel by the Lord Commander to become a Maester?
That's one reason for why he can't become a Lord (or hasn't become a lord yet) but that might not even be the case, the time is not at all right, he had also vowed to never be with a woman, but he broke that vow, I do not see why it will not be possible for him to break away totally from the Night's Watch vow after the Great War ends (if it does and the living wins.)
Spoilers à frente
Most of the answers above talks about his vows, totally understandable but until now Sam didn't even know his father and brother are dead, neither did Daenerys have any idea that Lord Randyll Tarly had one more son who could be the Lord. What about the Night's Watch vows? Oh, the Wall has fallen, the ancient order of the Night's Watch may not even be relevant now so how does the vows will matter after the War ends? If Jon Snow after becoming the King wants he can name him the Lord to at least save one great house from extinction, just like I believe he will legitimize Gendry to save the House Baratheon.
It is probably because of that scene with his father; when Sam was at home just before going to the citadel.
They were having a family dinner, and when Sam's father found out about Gilly being a wildling, the women left the table, and when they were alone, Randyll told Sam that Gilly and little Sam are welcome to stay there, Gilly will have to serve as a servant and he will raise his grandson, but Sam will no longer be welcome in Horn Hill and has to leave by first light. Sam then proceed to get Gilly and Heartsbane and go to the citadel.
So it is implied that Sam's father banished/disowned Sam, when he found out that his son is with a wildling.
While his vow to the Night's watch provides some technicality and may have mattered if this happened a few years before current events, the main reason is because naming a successor is não é uma prioridade agora.
Daenerys and Jon do not own the lands around Horn Hill right now, the Lannisters do and will certainly not give them to Jon's best friend. Furthermore, Jon and Dany are in a quite pressing war for basic survival (vs the dead) and another war for supremacy (vs Cersei). They simply do not have the time to appoint lords over contested lands right now, that is for when they win.
At that point, it might be Horn Hill goes to another family. Though if Dany/Jon prevail and Sam survives there is also a good chance he gets the land, because why keep the Night's watch after its purpose has been fulfilled?