Hey, I am a senior developer, worked in many companies. Now for obvious reasons, entered into the freelance arena.
From the description, it is not clear whether you need to create an API for your WordPress site or integrate an existing API (from the gaming site) to your WordPress site. In any case, I have done both. I am still working on a few such projects involving REST APIs for WordPress.
My bid is just to get your attention. Would like to connect and get more details about the project. Once we know our exact requirements, we can finalise the amount.