Are there parties other than the at-fault driver against whom I can take legal action in a Florida auto accident case?

You have the right to bring a claim against any party responsible for contributing to the cause of the accident. So, the answer is yes. If, for example, that driver had their car maintenanced at a repair facility, and it was negligently repaired, and that contributed to causing the accident, then you may have a claim against the repair facility, in addition to the at-fault driver. 

Moreover, if the driver is a separate person than the owner of the vehicle, you may also have the right to bring a claim against the owner of the vehicle. So, yes, you do have the right to bring a claim against parties other than the at-fault driver.