I am fairly new to PHP (about 8 months). I am building a web app, which is almost ready for beta. I am only now starting to think about what I would need to do to make a mob
No you do not need a framework to build a REST API, but it is highly recommended, as a well built framework can take care of things that can be very difficult and complicated otherwise, namely session authentications and caching and well separated architecture. Reinventing the wheel only gets you so far.
I'm a developer of Wave Framework which was developed keeping in mind an API-centric design (read more here). I encourage you to take a look into this framework and see if it might be something that could help you. It has a small group of developers, but it is slowly gaining recognition.
I encourage you to take a look at that and if it might fill your needs.