Officially, in florida, you are correct. Car Seat laws are governed by states and Florida doesn't care if you are in a taxi or your own car. On a side note, Illinois law requires them in taxis as well.
It does not mean that cabs are not an option. Just that it requires a little more leg work. Call the company and ask, and then call for a cab instead of just hailing one. They should be able to accommodate. It also means that if you choose to take a cab without one, you need to be ready to accept whatever consequences - regardless they are physics related in an accident or ticket related. My best guess would be that on site trips, they do not police much but accidents still happen. You have to make the calculated risk for your own family.