The first of September is just around the corner, but you can still save on preparing your child for school by using the delivery service of American online clothing stores.
From my personal experience, I can recommend two to you.
JCPenney
Here you can find both simple clothes for yourself and your child, and a department specifically focused on the school "uniform". I will note that the prices in comparison with Moscow are very moderate, while you get clothes from quite well—known, and sometimes very well-known brands at delicious prices.
Lands’end
This is a mono-branded store, that is, they produce all the goods under their own brand, which makes them think about quality. Believe me, it is as difficult to meet low-quality goods from them as it is to meet a virgin in the second year of a Moscow university. The products of this store may seem a little conservative to you, and they are. But for school they have almost everything and it's cheap and angry.
The issue of delivery deserves, of course, attention, but there is not much to say here: both stores deliver both by regular mail and by courier delivery. Taking into account the fact that the limit for customs—free importation in Russia is one thousand euros, you can choose any type of delivery without fear of delay at customs, in order to collect school clothes at their prices for a thousand or more — you need to try very hard.
Regular mail goes to Moscow, as a rule, about 10-15 days, courier 7-10. Therefore, I recommend to hurry up.