Yes, you can rent with a Florida driver's license. International licenses, to the best of my knowledge, while accepted in many countries are in reality only certified translations into several languages of the info on your 'real' driver's license.
For the larger companies like Avis & Budget, you can make a reservation in advance online. They operate much as they do in the U.S.
Many small, private rental companies around as well.
I was asked once to leave my passport with a company & refused. They said "okay" and gave me the car anyway.
Check the car out well with them for dents/dings & make sure they're marked on the rental form before you leave the agency. I think it was on DR1 that someone suggested actually taking digital photos of the front/back/sides of the car while at the rental agency. Not necessarily a bad idea if they attempt to blame you on return for damage that existed before you drove the car.
Now, on to the usual disclaimer: If this is your first trip to DR, you really shouldn't rent a car. Driving's a little chaotic and roads not always well marked. Use taxis, buses, or hire a driver. For your second or third trip, you may consider renting once you have a feel for how people drive and know your way around. Many never drive here.