Mt. Rushmore is almost surrounded by the Black Hills National Forest, with lots of inexpensive campgrounds. I have used Pactola, Roubaix, Horsethief Lake, and Hanna. All were quite nice.
At Yellowstone. There are several N.F. campgrounds just west of West Yellowstone (west entrance) and also just east of Cooke City (northeast entrance). On the Cooke City side, I like Chief Joseph, Fox Creek, and Crezy Creek. Fox Creek has electric hookups. The last that I heard, Chief Joseph. Colter, and Soda Butte were restricted to Hard sided campers, due to bear activity. I have not tried the camp grounds on the West side.
You can find a directory at Campground Directory and National Forest Campgrounds
. Check S. Dakota, Wyoming, and Montana. They are mostly all less than $15 per night except Fox Creek.
If you have a senior pass, they are all half price.