As said, if on property it's easiest to link a card to your room key- they bill everything in one big chunk at the end of your stay and you get an itemized list on your door the morning you check out. It saves a CC bill that could be quite long.
If not, then they still take CCs. I don't remember what all (the few times I needed to use a regular card, I used a MasterCard), but the big four (MasterCard, Visa, American Express, and Discover) are pretty much shoe-ins nowadays.
Disney is plenty happy to take your money and they try to make that process as simple as possible.