That's a good idea. Roll roofing would be even better. It comes in 3' wide rolls I believe.
Also, if you have an industrial park nearby, you can likely find some scrap plywood in their dumpsters. Large items sometimes come in plywood crates. Those usually get tossed in the dumpster.
Menards usually has a damaged section near the plywood and lumber. It's usually on an endcap of the isle closest to the lumber counter. There is usually paneling, and other types of sheeting that has been damaged, and sells for cheap.
I'd try to find some 3/4" plywood for the sides, then use...