You would need to go to a
Disney Store to recharge them, or wait until your trip and do it there.
Interestingly, you *can* buy the dining plan by calling WDTC, even if you don't have a package. Not many people here use the plan at all, so there aren't many people to use as examples, but an every-so-often poster did it last December-ish.
And gosh, why wouldn't they, right? It's just vouchers, and it's hard to really get more than your money's worth...
If you see a cart that's more permanent, that will likely take cards/giftcards. If you see one that can be pushed around, they won't have the capability. So take a little cash if you like your ice creams and such. Also, the ATMs at Disney are Chase, so if you have Chase or former-WAMU you're in luck!