Not to put a damper on it, but unless you get lucky it will be a chore. As mentioned, things ease up come April, but starting Feb 1st most everything will be booked. We do pick up days at County and State parks every year as we travel through Florida, but Monroe County (the Keys) does not permit camping in their County Parks, and the four State Parks fill up a year in advance. The only place I can think of might be the Kmart parking lot in Marathon. It is a beautiful trip but you may want to book a day or two near Key West if you can so you don't drive all 120 miles down and have to turn around and drive back.
Dave