We live in No.Va. too. Our kids are grown so we don't seek out the specifically Kiddie type places but we do have two favorite places that our son and grandson enjoyed.
1) Assateague National Seashore. Beach, fishing, swimming, biking, canoeing. Bikes. canoes and stuff can be rented in season. Good Ranger programs and a new nature center opening this year. No hook-ups, hot showers or camp store but good basic camping. We harvested shellfish and had a seafood feast. Grandson loves mussels. Go figure.
Camp on the bayside or the oceanside. And don't forget the wild horses. If you want hook-ups and hot showers the adjoining Maryland state park might be better for you. We've not stayed there because they did not permit dogs until last spring and if Bubba can't go we camp elsewhere.
2) The CGs on Skyline Drive are also favs of ours. Big Meadows is the best when camping with kids because of the more extensive programs, campfires, nature walks, films (think rain), great visitors center. Also has good camp stoe and a small restaurant. Again no hook-ups. Big Meadow does have 65 no reservation sites. Other CGs in park are all first come,first serve but have fewer programs.
Check them both out online. Some sites and some seasons you do need reservations. There are many nice state parks nearby and there are the usual KOA and Jellystone as well as privately run campgrounds. Enjoy!