12 Tool Proficiencies, 15 if you count musical instruments, 16 if you also include gaming sets.

Let's take this piece at a time.

Raça

Human with the Prodigy Feat, Envoy Warforged, Rock Gnome, or Dwarf. Nets +1 Tool Proficiencies.

fundo

Any House Agent background (Eberron), except Kundarak, Medani, or Tharashk (Because those grant Thieves Tools, which is a waste of a slot) Nets +2 Tool Proficiencies

Talentos

Skilled, obviously. +3 Tool Proficiencies. That's the only feat that grants Tool Proficiencies, and you cannot take it more than once.

Classes

Bard: 1 level IF you are counting musical instruments. They get 3 Musical Instrument proficiencies at level 1. This must be your First Class, or you only get 1 musical instrument

Rogue: 3 Levels, Mastermind. +3 Tool Proficiencies (Thieves' Tools, Disguise Kit, Forgery Kit) and +1 gaming set

Cleric: 1 level, Forge Domain. +1 Tool Proficiency (Smith's Tools) OR Knowledge Domain (Channel Divinity to gain temporary access to Proficiency with any 1 Tool)

Monk: 3 levels, Drunken Master. +1 Tool Proficiency (Brewer's supplies) OR Kensei (Calligrapher or Painter's Tools)

Fighter: 3 levels, Battlemaster. +1 Tool Proficiency (Your Choice)

Magias

There are no spells that grant temporary proficiency. The only magical effect that lets you 'fake' proficiency is found in the Knowledge Domain Cleric, which only lasts for 10 minutes (not long enough for a major project), but it's a choice between that one, and a Forge Domain cleric (which grants a permanent proficiency). So, do you want short term flexibility, or a full proficiency?

The maximum number of tool proficiencies is 17
If musical instruments do not count, this maximum drops by 2.
If gaming sets do not count, this maximum drops by 1.

O personagem

Connor is a human with the Dragonmark of Making (in Wayfinder's Guide to Eberron) and these classes:

• Bard 1 ------------------------- (for musical instruments)
• Cleric (Forge from XGtE¹) 1 --------------- (for smith's tools)
• Monk 4 ------------------------ (for 1 set of artisan tools or 1 musical instrument, 1 other set of tools, and the Prodigy feat from XGtE)
• Rogue (Mastermind from XGtE) 4 ------ (for thieves' tools, disguise kit, forgery kit, 1 gaming set, and the Skilled feat)
• Fighter (Battlemaster) 3 ---- (for 1 set of artisan tools)

^{1: XGtE refers to Xanathar's Guide to Everything}

As ferramentas

Note: If musical instruments count, bard must be chosen at level 1. If musical instruments hold less than half the weight of other tool proficiencies, then choose Druid instead for kit herbalism and multiclass into bard later for one musical instrument

Your Monastic Tradition can either be Drunken Master (for brewer's supplies) or Kensei (for painter's supplies or calligrapher's supplies), both from XGtE.

Your Roguish Archetype must be Mastermind unless gaming sets do not count in which case you can also choose Assassin (which replaces forgery kit com poisoner's kit)

Dragonmark of Making grants an artisan tool proficiency of your choice.

The Prodigy feat grants any tool proficiency of your choice, while the Skilled feat grants you 3 of your choice.

A customized background can grant any two proficiencies in tools according to the [Player's Handbook or Regras básicas.

To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds.

Final tally:

• 3 musical instruments of your choice -ou- kit herbalism and 1 musical instrument of your choice
• smith's tools
• disguise kit
• ferramentas de ladrões
• forgery kit and 1 gaming set of your choice -ou- poisoner's kit
• 3 sets of artisan tools of your choice -ou- 2 sets of artisan tools and 1 musical instrument of your choice
• 6 of any tool of your choice

Material extra

• As a Mark of Making human, you can add a d4 to every ability check with a set of artisan tools.
• As a Rogue you can choose to gain Expertise in your thieves' tools.

Temporary Proficiencies

If musical instruments do not count, an alternative build can acquire a different proficiency than kit herbalism, albeit temporarily, using the spell jarra mágica on an NPC in Waterdeep (spoilers):

Thorvin Twinbeard in Waterdeep - Dragon Heist

If you forgo the level in Cleric and only get 3 levels in Monk and Rogue, you can replace Bard 1 with Wizard 11 to get jarra mágica which will both replace the smith's tools you would've gotten from cleric as well as grant you mason's tools. (You then use the wizard's Ability Score Increases to get the feats required).