You do need to call to make a reservation. Groups must be 15 or less, or be split for control. As for places to stay, there is one camp near Murray which sometimes takes Scout groups. The best place to camp is at Land Between the Lakes, just East of the Museum. Groups can camp in the public campgrounds.
Fees are a bit high, but probably worth it.
Energy Lake Camp is for groups and will take reservations. In the past, there has been a very active program for Scouts, including offering Merit Badges. Reservations are not needed in the public campgrounds. The larger ones don't fill up except holiday weekends. There are a few motels in Murray.