If you don't want to fight through all the crowds in the big supermarket in Le Mans, there are a few smaller ones not far away. There I 2 I use in particular. One is right at Mulsanne Corner so you can shop then do some spectating (or vice versa:D ). If you fancy going a bit further (5-10Km), Head out of Le Mans on the D304, as you come into the first village there is a nice supermarket on the Right.

If you fancy eating out, and want to try a typical local french restaurant I would recommend going further along the D304 to a small town called Le Grand Luce (its about 15 miles from the circuit) There is a small restaurant called Le Normandy in a little square. Good simple french food and about 10 for a 3 course set menu.

Hope this helps.