Your "key to the kingdom" is connected to a charge card of your choice, usually the one you booked your room with. But they will ask to confirm that upon checkin.
You can choose which members of your family get charging priveleges and which ones don't. You can also put a limit on how much you can charge. I believe the default limit is $1000 per day per room unless you request otherwise.
So, for example, if you booked your room with an American Express card, that same card is connected to your room key and anything you charge on that card throughout the hotel and parks is charged to that same American Express card.
I always carry at least $20 in cash just to be safe. And don't let the convenience of the card get to you - things like pins REALLY add up when you're not paying with green. On my last trip I spent well over $100 on Disney pins without really realizing it until I looked at the bill. Sometimes things just don't register - 10 small $10 purchases over the course of 2 days add up to be more than you may have wanted to spend.