If you don't want to carry cash, get a gift card in the amount of your dining budget. We pay for our trips entirely with gift cards and I wouldn't have it any other way. Plus, if you come in under budget on your food, you can use the remainder on souvenirs (or whatever else strikes your fancy.)
The only thing you can't use any sort of plastic on (gift card, dining plan, KTTW charge, credit card) are roving vendors, like those who sell light up stuff during the parade. I can honestly say that although I carry a very small amount of cash (along with ID, ticket, and room card,) I have never had to use it, and I have spent a ton of days there in the last couple of years.