Need overnight near Danville VA

If its not too far out of your way you could swing down into N. Carolina and stay at Tanglewood Park in Clemmons just south of Winston Salem. Very nice. But it will be at least an hour and half drive to the race track.

Philpott and Fairy Stone are the closest decent ones to 58, a little north and east would be Smith Mountain Lake State Park
SML State Park.

Edit add. Forgot all about Hyco Lake park. Thats a small campground but likely only about 9 miles southeast of Alton. Never been to that one but have visited the other run by Person County and it's clean and well maintained.

Fairystone state park, Philpott COE, Salthouse branch COE

This is what Campground Reviews (formerly RV Park Reviews) lists as within 25 miles of Danville. Slim pickings, IMHO, and some not very recent reviews. If you were coming from the east or north, I’d suggest checking the Corps of Engineers campgrounds on Kerr Lake Resevoir, but that’s about 75 miles east of Danville. There are some COE parks about 50 miles northwest of Danville on Philpott Lake. Haven’t been there, but they are rated pretty well.

Try zooming in on the this map of campgrounds in VA to see what's nearby.
Allstays Map of Virginia campgrounds.

Wal Mart?