Co-worker went last year, with his 30 ft Avion TT and Loooong Dodge 3500, and stayed at Center Lake Campground. It is "same day reservations" basically walk up only, so there will be sites open even in the busiest times. No electricity. $19 per night, plus a $20 State Park fee good for 1-7 days. Vault toilets and showers.
http://gfp.sd.gov/state-parks/directory/custer/campgrounds/center-lake/Just stay off of Iron Mountain Rd and Needles Highway with a Motorhome or trailer, due to tight tunnels and switchbacks.
Also be aware of the total solar eclipse on August 21st, and its path. Campgrounds and such are very booked up along the path of it and either side of the path, so plan your travels carefully.
http://xjubier.free.fr/en/site_pages/solar_eclipses/TSE_2017_GoogleMapFull.htmlCharles