Ahoy, Stacey!
Since my Father-in-law wears an insulin pump, I called him to ask for his expertise. He informed me that his pump can usually only be disconnected for about 45 minutes and that it would need to stay cool while he was not wearing it.
Unfortunately, on
Castaway Cay, there really isn't somewhere official where you could keep the pump. However, you could bring a small thermal bag and a reusable ice pack that you could keep cold in your stateroom refrigerator. Then, you could bring these things with you onto Disney's Private Island, and leave them near your beach chair while you snorkel. There are plenty of beach chairs that have umbrellas, so your bag could be in the shade.
Our family always brings a beach bag onto Castaway Cay with us, and we have never had an issue with our things being disturbed while we are swimming or snorkeling. I don't tend to have anything too valuable in the bag, but our Key To The World cards on our lanyards and our phones are usually in the bag, along with sunscreen and goggles.
I hope this helps with your cruise planning. Please come back and visit us here at the Disney Parks Moms Panel with more questions as your cruise gets closer!
Happy Cruising, Tracy V.