Its a national park so there should be a website. Sites do not have any RV services but there is a dumpstation and the bathrooms should have showers. I have not been there but I intend to check it out soon so I can provide more reliable feedback to members. Check with the rangers, probably by phone, to discuss transportation into DC. Alternatives are Pohick Bay and Bull Run Regional Parks in VA, both have electric sites and a dumpstation. Or Prince William Forest NP near Quantico, which has no services for RV. There is a concession-run RV park associated with Prince William that offers full services and will also allow dumping at a $5 fee or so. Problem is none of the alternatives appear to have any transport to a subway stop. If you have a toad, you could drive to a subway station for any of these.
We are going to the DC area in a couple of weeks and after a lot of research I ended up booking at the Fredericksburg KOA in Virginia. It is about 1 hour from Washington, but they have a shuttle that will take you to a metro. Although it's a bit of a hike the campground has good reviews and it's got all of the hook-ups. As mentioned, if you don't mind no water hook-ups, Pohick Bay in Lorton looked like a nice place and was much closer.
Don't know anything about Greenbelt park, but spouse and I spent two weeks at Cherry Hill Park in College Park, MD. Great park, but pricey (we paid $45 per night for Full Hookups). Park is convenient to the bus and metro into DC, has many amenities, including wireless.
Unless you have a specific reason to be in Maryland, I'd say camp in Virginia. Bull Run Park (near the Manassas battlefield) is within a 15 minute drive of METRO and 35 minutes (non-rush hour) of the mall. Even though you are in the middle of a built-up metropolis, the sites are large, privacy is excellent, and the rates are cheap. We stayed there for nine nights last year and we found it to be quiet; I can't comment on how it was during daylight hours as we were doing the tourist thing. Camp staff was probably the friendliest we have ever encountered, and since there is a major shopping area within 10 minutes drive, including 3 grocery stores, if you forget something it's no big deal. Their web site is here. Bull Run Park is run by the same folks as Pohick Bay.