I\'m planning to write a 3D FPS game, based on WebGL.
Yes you should definitely use a WebGL framework, it will save you a lot of work.
The WebGL ecosystem is in such a rapid state of change at the moment that it is quite difficult to choose a particular library. However, of all the libraries I have seen, I would suggest Copperlicht as it seems to be aimed at FPS games in particular. There are so few fully-developed WebGL games published right now that it is difficult to say what library is popular.