If I was hit by a car while on my bicycle in Florida and the car had no insurance, can I still recover?

Yes, likely you could still recover if you were hit by a car that had no insurance. Firstly you could potentially sue the driver of the car personally. Also there could be other parties to the accident who have some responsibility, who could also be held responsible and be forced to pay for some or all of your damages. Not to mention you could also call upon your own automobile insurance or potentially your homeowner's insurance, depending on how those policies are written, to cover any of the damages that you have.