I've seen several sites out there that list FBO's state by state but they never seem to be complete. If you want to try a bit of 'backwards engineering' try this link
http://www.airnav.com/airports/us/FL
You'll have to got through each airport in turn but as far as I can tell it's a fairly comprehensive list, complete with lots of info you never really wanted to know. (good site though IMHO).