I looked in to the documentation and searched for 1 hour i think and i did not find an answer

the documentation did not help at all so i decided to go here to ask for help

how do i check if the current user owns the game with the itch.io api to prevent running it without purchasing

You can't (normally)

Itch.io only distribute DRM-free games

If you have an API key, you can check if the play purchased a game