In Sokolniki park, highly recommended if they open up this season: www.sokolniki.com/eng/Services/Camping.aspx. In English!
A compilation of resources in campers and equipped campgrounds: www.autocamper.ru
A campground just north-east from Moscow: www.buhta-radosti.ru, + 7 916 117-3313.
A campground near Sergiev Posad, on the Torbeyevo lake: www.torbeevo.ru, +7 495 857-3611.
All but Sokolniki cater the the local market and may not be able to easily interact with non Russian speakers.
Just a reminder that a camper can be rented from me by qualified clients. I can also arrange camping trips around Gold Ring or between Moscow and St. Petersburg.