If I maintain an open source project, could I build a doc page using a small GPT allowing users to query FAQ and common methods using natural language?
You can build anything using numpy. You can build a supercomputer out of duct tape if you want to.
Spinning up a db, serving an API, doing natural language processing, ... whatever you want.
That said, there are niche solutions that do these things well and that can save you a lot of work. There are also frameworks (such as Pyntango/Jupyter/Nbspectrum) that lets you spin up a simple API quickly.