I used my Credit/Debit card when we ate a Casey with no problem.
When you check in you will need to give them a credit card so you can charge off of your room key.
All of the WDW areas are very good about taking credit/debit cards not only do the shops,TS and CS accept them but even all the food carts throughout the parks as well.
I think the reason pretty much all of the carts are able to accept credit/debit cards is because they all have to have the capability to scan tickets for the dining plan so that people could use snack options.