Going to be taking a serious look here;
First issue I have is you seem to be mixing up Patrons and Pact Boons on Warlock; for example, the Watchers would be a Great Old One Patron, but they may supply you with ancient texts inscribed in a language only you, the Voidspawn or other Void-touched beings can understand (Tome). Another case would be something like Rhaast would make an amazing Sentient Weapon Patron for the Xanathar's subclass on Warlock, the Hexblade. __You removed the Otherworldly Patron at Level 1, which I would not advise; the core concept of a Warlock is that their Patron gives them their powers, whether or not it is intentional.__ This can take effect as deals with Demons and Devils, or as being touched by creatures who's own state of mind is alien in nature. In essence, Warlocks get their Subclass at Level 1, much like how a Sorcerer or Cleric does.
Examples of a Patron in Runeterra could include the Darkin, such as Rhaast or Varus (though that is VERY DM discretion), Demons such as Evelynn or Tahm Kentch, or Spirits of the Shadow Isles, namely Kalista or Hecarim.
Also Pix. *Pix is cool.*
A few races that could be very entertaining to play are Yordles, Hexforged *(Warforged, though I'd advise looking at UA 5E content for that rather than Ebberon, as it has a little bit more of an interesting take in my opinion)*, and Vastaya. I personally wouldn't advise making Vastayashai'Rei playable (Shapechangers are a fucking meme and a half for DMs btw), but you could easily set it in the past before Demacia existed. Maybe predating the Rune Wars, even. *Hell, I'd love to do a campaign around bringing down Mordekaiser. ~~Focus Blue, FOCUS!~~* As for Human, I personally think that keeping it as 5E PHB standard works well enough; +1 to All or +1 to two, Skill and Feat is very freaking strong ~~and why Human Fighter is the most frequently played combo. You can do literally anything with that, in terms of builds. Level 1 Shield Mastery? Why not. Level 1 Tough? Screw it, HP out the motherhunker.~~
I feel that Yordles should have Persuasion and Deception as Skills from Racials, as well as possibly a +2 Cha +1 Dex? I feel that may be acceptably functional. Small, obviously, 25 Feet Movement because Small, 60 Feet Darkvision probably? I'd also give them A Touch of Glade (Can Cast Disguise Self for a Small or Medium Creature, but still classes as Small in terms of size, etc etc three times a Long Rest)
Vastaya are probably going to be the biggest trouble childs in this, given how unbelievably varied they are. You've got bird Vastaya, you've got fish Vastaya, you've got cat Vastaya, you've got goddamn otter Vastaya. **Otter. Vastaya.** Classification would probably be vital here; Avian, Feline, Canine, Aquatic and Mammalian probably would be the best go to's. I'll link a file on my thoughts for each.
Vastayashai'Rei are... interesting and damn unique, in that they can shapechange regardless of size, it seems, though I'd argue to restrict it up to Large Creatures, since Neeko - the only known Vastayashai'Rei - has been known to shapechange into the form of a Bird; Druids' Wild Shape on steroids, I think?
Note; I didn't scroll all the way down.
*I want to fucking die.*
A general note on Backgrounds is that they will either give you a Choice of 2 from a list of Skills, or will give 2 Skills. You seem to be under-powering it on that side of things, since you give only a list of 3 Skills and allow for a Choice of 1 from the list. They also tend to give a little bit of starting Gold and a few other items, such as a ring signifying your house crest or a hunting trophy.
Demacia seems somewhat okay, though a free defensive capability isn't necessarily the best option here; I'd personally give Demacians Medium & Light Armor Proficency with Shields, as well as Weapons Training for Rapiers, Shortswords and Longswords, as well as Shortbows and Longbows. Perhaps it might be better to allow them to simply use up to 3 Hitdie in combat once per Long Rest? Depending on Class, that can be *A LOT* of outright healing, especially on Fighter or Barbarian, while Casters might need the HP just to survive a particularly bad attack.
Noxian Diplomacy; **Not a good idea.** 4D12 potential bonus damage on a Longbow for a character that has Sharpshooter? That will break your damn game wide open, since the Ranger that could be would be dealing 4D6+1D8+5, possibly a +8 at the end if it's a +3 Longbow. **That attack's average damage from Die alone will be 26 Piercing, assuming you only roll Average. That is against a target that may have only been punched once. Watch what happens when a numberphile Human Rogue Assassin one-trick sees this and gets to play it.** *This assumes IT HAS NOT STACKED YET.* On top of the fact that this is regardless of class? *Laughs in Rogue Assassin 14th Level Way of the Shadow Monk 6th Level.*
I would have argued that Noxus should give Weapon Training for Handaxes, Greataxes, Battleaxes, Warhammers, Longswords, Shortswords and Greatswords. Maybe also Shields. What you currently have down would make a UA Tweaked Ranger from Noxus potentially end up dealing 2D8+4D12+ Dex Mod and Enchant to a single target, which is freaking stupid in terms of single hit damage. And god help that target it it *CRITS.* **I personally would HEAVILY advise AGAINST this.**
Piltovans don't necessarily make weapons for themselves; some might, while others might make goggles that would assist in seeing from a greater distance, or a pack with all the tools one might ever need. I personally would advise to also clean it up a little, since 'Magic Damage' doesn't say much in D&D. Magical Piercing? Force Damage? Elemental or Magically Enhanced Phys?
Also, I saw Ixtal. *~~Still think they would've done better as a dead culture...~~*
Truth be told, I'm not that well informed on their lore ~~mostly because there was no real reason to add them imo other than having a totally-not-MCU-Wakanda Region~~ but the effects seem... strong. That could be a bit too much as well; might be a good idea to *lightly* tune it back.
I did see the Mageseeker component, and I feel like it would be a better alternative to use the Mageseeker as a Demacian Background Variant, much like how Noble has Knight Variant. There are two primary reasons for this; First of all, as far as I am aware, Mageseekers themselves generally aren't Mages. The only exception that I can think of was Sylas when he helped them, as he innately could sense magical energies. Rather, and as my second point, Mageseekers make use of captured magical capabilities inside their Petricite devices, and are often known as borderline exorcists, given how they seem to have the ability to 'cleanse' Mages of their powers for a time using Petricite. For example, a Mageseeker could use a Petricite Staff to capture an Eldrich Blast's effect when hit and simply neutralize it as it strikes. (Grant an AC Bonus against Ranged Spell Attacks and Reduce Magical Damage to Self, maybe? Though don't go too high on the DR, or you run the risk of it being too strong of a build; keep it balanced, keep it fun)
Also, on the Hemomancer, I would actually like the concept to be more that of a Blood Caster; make it so it's instead a Warlock Pact made specifically with Vladmir's cults in Noxus, and you give them some method of in-combat regeneration for 'lockslots. (perhaps dealing the Warlock Spell Level of Damage to self to regain a Slot? Early on this will be very dangerous and glass cannony, though, and might suffer the same issue as the old Blood Hunter variant did)
Also, though the Blood Pool is a good thought to put in - emergency escape tool and all - Misty Step exists, and if a Warlock is to deadass tell me they don't have it, then they deserve to get crushed to death by that boulder they could have Misty Step'd away from.
A note on the Augs from Zaun and Piltover; **do not make them Subclass-only.** Argueably that sort of technology should only be coming from Artificers from that region (see Eberron 5E UA if you don't have the official book yet), which would make sense, but do NOT bind it to a Subclass, as that sort of hardware is usable to any living creature.
Also, there are some Variant Rules from DMG, such as Sanity or Honor. The concept is that they both serve to act as unique saves whenever the DM calls for them; for example, seeing a thousand bodies raising up from the sands of Shurima in the middle of the night near a Void-touched ruin would probably be just *slightly* panicking, making you need to pass a Sanity Save. *Or just encountering Void Portals, really. ~~DON'T LOOK INTO THE LIGHTS.~~* Meanwhile in the case of Honor, I do *believe* its Save is used whenever you are forced to commit something that goes against your character's morals, such as killing another person in the case of a pacifist or perhaps defying Noxus in the case of witnessing Singed doing not nice things to Ionia.
Edit - 02/23/2020 - Continuation and Clarification, as well as minor patch-up.
Edit - 02/29/2020 - Cleaning up text that, grammatically speaking, is braindead. Also happy leap year day.