The best I’ve seen is www.put-and-go.ru. Outlets all over Moscow.
What you can normally find here are old-style Russian variety chains that are a pain to put on, eat up the tire in no time, and need to be taken off before even a short stretch of bare asphalt. The alternative is real low-quality products that, if you are lucky, will serve once. Both are to be avoided.
(1) By the entrance to VDNKh. Cheap Chinese but good enough for bike rental, which is the main product of this establishment.
An unconfirmed report of a shop that specializes in old Sputniks and such. Sokolnicheskaya ploshad 9 (in the big bike market I think)or at Tretye Pradilnaya 17 (Metro Semenovskaya?). Their site is www.velo-delo.com but at the moment it appears to be down.