Judy and Jerry,
We did this trip in Sept. 2010 but we did tow a car. We stayed at Broken Arrow Campground in Custer since it was a Passport America park for $15 a night full hookup. Nice park, but they do cater to horse owners. It was not a problem for us. GREAT location!
We did Needles Highway, Mt. Rushmore, Crazy Horse, Jewel and Wind Caves NP, Hill City, Mammoth Site, Custer State Park.
There is a campground shown online called Gulch Campground that gets good reviews that also has car rentals. Info is at www.blackhillsrv.net. We have never stayed there but I did notice they have car rentals. You can check www.rvparkreviews.com
for other recommendations on campgrounds in the area. We found Custer to be the perfect location for a base camp for sight seeing in the area!