In another thread, I asked if the FAA keeps track of FBOs. I
decided to submit that question to the FAA. I received a timely
and complete response.
You asked if FAA maintains a list of fixed base operators for each airport.

The FAA does not. You need to contact the individual airports for this
information.

It surprises me a bit that they don't know the FBOs are at an
airport. Perhaps some other organization (Transportation?) does?
I would expect some licensing of fuel facilities.

--kyler