In Moscow, you can pick up basics for cooking Japanese food at home at most higher-end grocery stores like the Bazaar at the Tsvetnoy mall and Azbuka. More specific imported Japanese groceries, drinks, sweets, and household goods are also easy to get at specialty stores.
Often, companies in Vladivostok import the goods from Japan and send them on to Moscow. The lucky residents of Vladivostok have an easier time shopping for Japanese products and food than we do here in Moscow!
To answer your question, I stopped by a shop called Nippon at 7 Tsvetnoy Boulevard. For a small store, there is a wide assortment of both food and household products. In their freezers, I found imported (Chinese) edamame, flying fish roe, and wonton wrappers, as well as local fish cut for sushi-making.
There are ingredients for miso soup, curries, soups, and sauces, as well as sweets, drinks, and vitamins. The back of the store has beauty products, cosmetics, and cleaning products. In the front, I even found coconut sugar, Japanese peanut butter, and gum.
Nippon also has a second location at 4 Kravchenko Street and an online store that delivers.
When I threw a Chinese New Year party one year, I used an online store called Yakitoria for some basics.
I will be doing a follow up about where to get sushi-suitable fish, meaning fish I haven’t had a problem with eating raw in Moscow.
Edit: a colleague of mine also recommends a store called Megumi for excellent Japanese cosmetics and detergents.