Amazon may be a last resort...but there is a way to reduce the use of plastic mailers from them as well. It involves a quick chat (through the "help" button at the bottom of the page). When I requested plastic-free packaging on all future orders, the representative responded in a very encouraging way + the request was noted on our account.
A couple of fun ways to use the cardboard that does come through might be to make some bears out of them + contribute to your neighborhood's bear hunt or to design the house of your dreams. :)