We stay in Custer State Park when we visit the Black Hills. http://gfp.sd.gov/state-parks/directory/custer/ There are several campsites within and outside the park, and in recent years some of the campgrounds have added electricity. However, currently the only dump stations are located at "Game Lodge" campground and one other site near the park volunteer campsites on the wildlife loop road. None of the campgrounds have septic hook-ups (that I am aware of). "Sylvan Lake" campground is located up the Needles Hwy., and would not be suitable for large trucks/campers/ RVs due to the roadway. Our favorite of the campgrounds is "Blue Bell", which is easy access. It is shaded and centrally located, and in addition to electric sites, it has an updated bathhouse, as well as newer restrooms and cabins. There is a decent restaurant, a small general store, and a horse riding stable a stone's throw away at the Blue Bell Lodge.Has anyone camped near Mount Rushmore? We're considering a trip there in the Fall so looking for any tips.
Campsites in Custer State Park are available by reservation only, and do book up quickly quite early in the season. So it's hard to say if there would be any availability for this Fall. Even so, whether you get to camp there or not, Custer State Park is well worth a visit. (The park entry pass is good for the week).
Other must-see's in the area are Wind Cave, Jewel Cave, the Badlands, and Devil's Tower; and the drive through Spearfish Canyon is beautiful.
Custer State Park campgrounds are dog friendly, as are the Black Hills National Forest, and the George S. Mickelson Trail. If hiking with your dogs in the National Forest, be alert for mountain lions. Chances are you will never see the big cats, but domestic animals can be attractive lures.
One year we camped in late September/early October for the annual Buffalo Round-up, and the weather was very comfortable, with only one day with a spit of snow in the air. Landscapes vary from the Black Hills to the Badlands, so pack for a variety of weather conditions.
Enjoy your trip! You'll find that South Dakota is someplace you will want to return to again and again.