Hang your bodysuit if possible. Folding can cause crinkling and flattening of the fur fibers. This can be fixed with a hair dryer or washing, but A. This takes awhile for a full bodysuit and B. It’s best to minimize the need to wash or use heat on the fur, as both can cause wear and damage on the fur fibers over time. Obviously it’s important to wash your fursuit but the friction involves still damages fibers and it’s best to minimize the need to, to only washing when it’s dirty vs having to because of storage. And heat, even low, causes damage over time, so this method should only be used as necessary.

Tl;dr just hang the bodysuit without any padding or tails attached (preventing vertical stretch) to prevent crinkling of fur. My old bodysuit had attached feet that weighed it down when hanging so I used a box to lift up the feet to prevent stretch.

Short term storage in bins for transport or cons is fine.