Yep, everything on the ship (except for future cruises and placeholders) gets charged to your room account, which you can then pay off using gift cards.
On Castaway Cay, Disney-owned shops and bars use room charges. However, the post office requires cash. I'm not sure about the Bahamian shops.
GET UP TO A $1000 SHIPBOARD CREDIT AND AN EXCLUSIVE GIFT!
If you make your Disney Cruise Line reservation with Dreams Unlimited Travel you’ll receive these incredible shipboard credits to spend on your cruise!