You could try this, probably 30 min drive to any kind of store. First come, first serve, no reservations.
Sage Creek Campground
Bison often wander through this primitive campground, located on the west side of the park's North Unit, near the Badlands Wilderness Area. Access is located off of the Sage Creek Rim Road, an unpaved road that may temporarily close after winter storms and spring rains. The road provides limited turnarounds for large recreational vehicles. Camping is free of charge. Pit toilets and covered picnic tables are available, but no water is available on-site. Sites at the Sage Creek Campground are available on a first-come, first-served basis and rarely fill to capacity. A portion of the Sage Creek Campground is designated for horse use.
My roughing it plan would be a cheap hotel with lousy wifi. In that spirit, Black Hills State U rents out dorm rooms by the week.