Hands down, the best campground in St. Pete would be Fort De Soto, which is a Pinellas County park. The campground is not directly on the beach, but there are many sites with water access for kayaks or canoes. The beaches, including a dog beach, are only a five minute drive to some of the best beaches on the coast. There's also the historic fort which is fun to stomp around and has great views. You can see some pics in my album: Forest River Forums - dkb74's Album: 2013 Camping in our 2014 Rockwood Mini-Lite 2306 - Picture
Reservations are hard to get. They can be made 6 months in advance.
Other than that, all you will find are various RV parks and a KOA, all off the beach.