I'd recommend shying away from plain vanilla PHP unless you for sure, for sure know what you are doing and you have a damn good reason for it. (and even then, you probably shouldn't but if you insist...)
It is a rite of passage for every PHP dev to think they can write their own framework. You may build something that works but you don't have the power of the global open source community out there. Feel free to do it for learning reasons but don't put it in production.
If I were forced to, I'd probably still use composer to pull in symfony/http-foundation and other components to help out like Doctrine for API Models/Repositories.
You can find an example of a [https://github.com/php-vegas/starter-app](simple php site) and use bits of it for an API app instead.
There are a few "Quick RESTful PHP API" tutorials out there but honestly, I'd advise from a security and maintenance standpoint you just pick any major open-source framework that either your team or you are proficient in or end up enjoying. It will save you a ton of time later when other devs or the product owner throws big feature requests at you.
Here are some other links I've found that seem promising but again, you're re-inventing the wheel with so many good frameworks out there... Educational purposes only.