They do accept American currency, but I suggest that you bring smaller bills if you plan on paying for items in cash. I bought something in Grand Cayman, and was given their currency as change. (Fortunately, it was just their version of our pennies.) I wouldn't have been thrilled if I was given back paper change.
While merchandise is reasonable, beware the exchange rate in Grand Cayman when it comes to food. We stopped for drinks/appetizers, and the price was quite steep when converted to our currency.
If you charge items, I suggest that you stick to the Disney recommended stores. Just an FYI, they convert the charge amount to pesos while in Cozumel.