You should try to use the materials as soon as possible but I've successfully stored strand cane, cane webbing, Oriental seagrass and natural pre-twisted rush for years without any problems either in my house or out in my storage unit.
You MUST make sure there's NO moisture in the containers and try to use either paper shoeboxes, paper grocery bags or large paper storage cubes, if you have a choice, so any moisture that might be there will be absorbed before causing mildew and mold.
When I first started in the business in 1975, I would coil up each individual strand of cane, tie with a twist-tie, and line them up neatly in a paper shoebox that was marked on the outside with the gauge of cane within the box. I stored my strand cane in this manner FOR YEARS!😜
Then, I decided tying each individual strand was just too much bother. 😳
So then, started to store all the cane bunches in large plastic totes and that's the way I've stored strand cane, cane webbing, and natural pre-twisted rush and Oriental seagrass ever since. 😉