Reims seems like a potentially good place so don't be shy to explore! The Decathlon has an accessible dumpster (with things in it of course), and some other big stores (like the toy store) at the same gigantic shopping area seem to have accessible bins. Look for the Decathlon at the south end of the city.
There is a Carrefour city at 131 Rue de Vesle (in the center of the city) puts their bin out, and one might also find fruit in boxes around it. (Last checked: Sept.2014)