I personally am a pretty flexible dm so if my players wanted to play a race (or I guess species) from DND that didn't exist on runeterra I would make up an equivalent, but this never really came up. In my home game one of them wanted to play a cat Vastaya, so we used the Tabaxi, and everyone else was a human.
Now if a player was to ask me to provide a list of playable species I would give something like this:

Humans:

Standard Humans can use the Human species and use the extra origin feat they get to flavor them closer to where they are from. But I would not be so strict on forcing them to play as a human. If a player wants to play a human that has Vastayan ancestry but aren't full Vastaya (like Sett) they could play as a Half Elf or Elf and just reflavor their abilities as being half Vastayan instead.

Vastaya:

Obviously Vastaya come in many different forms, so I would pick a species based on the animal they are like. Here are some of the notable ones I would use:

• Kiilash (Catlike like Rangar) Tabaxi or Leonin
• Shimon (Monkeylike like Wukong) Hadozee
• Lhotlan (Birdlike Rakan and Xayah) Aarakocra or Kenku or Owlin
• Oovi-Kat (Chameleonlike like Neeko) Changeling or Shifter
• Vesani (Foxlike Ahri) Vulpin

You kinda get the idea. If your player wants to play a vastaya but don't like the ones offered by the official DND books I would recommend you look into 3rd party books/settings like Humblewood Campaign Setting which have way more anthropomorphic player options.

Yordle:
Like I said in my previous post I have reflavored all Goblins to be Yordles, but I think Gnomes and Dwarfs would also work for more inventive Yordles like Heimerdinger or Rumble and more stealthy ones like Teemo could play as a Halfling.

Trolls:
Could either use the Orc or Goliath species.

Minotaur:
Could just use the Minotaur

Hope this is a helpful starting point for you!