Kimiko - just restocked and had a long discussion with an employee at The Caning Shop (Berkeley) who also happens to be big into SCA activities. He said you could easily spray down the reed and stretch it, so I'd imagine you could do the same with dried out stock. He wasn't sure how it would wash up inside a garment (washed by hand) - do you have experience with that?
Thus, I keep mine in fabric-covered cardboard boxes just large enough to handle the rolls. I would think putting them in ziplock bags they'd actually fare well since that's how I keep rubber bands from getting dried out. It's buttons we want to keep in breathable containers because of the off-gassing - not sure reed has that issue.