One family in our group is going to turn in a rental car at the Florence airport at the end of one day, then fly out at 6am the following morning. They are having a problem finding transportation at that early hour, as many shuttle buses, buses, etc. don't start operation until 7am. They're considering a taxi, a little apprehensive that they will get there at the early hour.
Generally the hotel should be your first source for solving that problem.
As Frank says, have them enlist the assistance of their hotel. We had a 7am flight and a day or two before, asked our hotel to arrange a taxi. Not a problem.
wherever you are staying thay will arrange for a taxi and I assure you they will be there waiting!