I've actually been looking at those plastic deck storage containers to try to fit a bunch of kite into. They don't seem to make them in anything over 56" long though.
Okay, confession: I leave tons of things in totes-- things I want to last a long long time. The stuff I've been yammering about is an archival thing and it has more to do with preserving precious art. If you have a collection of kites.... like a historical collection and you mean to see they remain pristine for the next 100 years, you'll want to look into the plastic-thing. Too, not all plastics are created equal-- PVC and acetates are the bad guys.
If you don't see any discoloration or actual rot, it's likely that things are just fine. Change out the cardboard for a sheet or something.
Chilese's ideas about "tight is bad" is really good. Air circulation will help prevent moisture from doing its bizarro thing. Darkness will prevent fading and rot from the light.