SHORT ANSWER
No, you don’t need a framework to achieve your goal.
BUT
it will be really easier if you use a framework to manage your API. I suggest you to go for a lightweight framework and maybe you can convert easily your webapp to the framework too, having one “app” to return two different “things” (web stuff & API).
Take a look at Laravel, Laravel 4 based REST API or a list of popular php rest api frameworks that can be used to build one.