http://www.hoover.org/library-archives/collections/soviet-union

The Hoover Institution has an extensive archive collection.

There are several websites with Soviet archive materials.

German Documents in Russia contains the contents of the Central Archives of the Ministry of Defense's Fond 500. It is a joint project between Germany and Russia to digitize documents captured in WWI and WWII.

Interbrigades of the Spanish Civil War contains the contents of the Russian State Socio-Political Archive's Fond 545. As expected, these are documents of International Brigades fighting in the Spanish Civil War.

USSR and the Allies is the Ministry of Foreign Affairs' archive, regarding diplomacy between the Allied nations during WWII.

The State Catalogue of the Museum Fond of the Russian Federation contains a great number of artefacts, some of them documents, from Russian museums.

Documents of the Soviet Epoch contains GOKO decrees and Stalin's personal library. Note: you will need an account from Yale University Press to access this archive from outside of Russia and Belarus.

Hardened by War a selection of archive documents from Soviet archives.

Heroism of the People award orders for Red Army soldiers.

Electronic Archive of the Soviet Military Administration in Germany contains documents from the State Archive of the Russian Federation connected with the post-WWII occupation force in Germany. Note: this site is terrible and frustrating to use.

Memory of the People a vast collection of maps and journals kept by Red Army units, very well organized and searchable.

Additionally, many Russian historians post scans of documents on their blogs. My favourite is Yuri Pasholok, but there are more. There's also my blog where I post and translate Soviet documents pertaining to armoured warfare.