In central VA, look into Walnut Hills outside Staunton. I've stayed there and it was OK. Roads are little tight, but the park is pleasant and it's a short hike to Staunton. I've also heard good things about Misty Mountain Campground in Crozet but I've not been there yet. Both these have web sites.
Grayson Highlands State Park is one of my favorite places in Virginia There's just so much beautiful hiking, and the Virginia Creeper is nearby. It IS hard to recommend a campground though without knowing what you like to do. Grayson is really remote; no cell service; and not alot to do that isn't related to nature.