First that comes to mind is the North Chick Cumberland Trail area. There are a couple nice camp sites about 4.5 - 5.5 miles in. The area can be infested with rednecks heading to the blue hole with glass beer bottles, so if you decide to head in here, please bring a trash bag with you and clean up along the way. This is a beautiful area, and with a little help from concerned hikers/campers/runners, we can reclaim the area.
Chester Frost Park is the closest camping I know of to Soddy Daisy, if you are looking for car camping. They do allow RV's but there are tent sites away from them. Some of the tent sites are right on the lake.