I did some research and posted it in a thread a couple of years ago, back when I lived there. IIRC the best bet was the National Park Service campground at MD Rt 193 and the Beltway, but it's dry camping. A short but dangerous hike to a Metro station.
You could call Cherry Hill RV Park on Cherry Hill Road in College Park, but they look pretty snooty from the road; may have an attitude about skoolies.
Don't know the VA side of things.
I do not suffer from mental illness. I enjoy every damn minute of it!