As in other Mexican resort areas, Cozumel's shops accept dollars as readily as pesos, and many goods are priced in dollars. You'll get a better price everywhere on Cozumel if you pay with cash or traveler's checks since there is a hefty surcharge on credit cards that you may be asked to pay. MasterCard and Visa are the two most popular cards, followed by American Express. The many warnings you will receive about buying from street vendors should be heeded -- the quality of their merchandise leaves much to be desired, although this may not be apparent until it's too late.
Cruise ships usually dock at Cozumel on Monday, but there is traffic here almost every weekday, and the shops are the most crowded 10-11 and 1-2. Traditionally, stores are open 9-1 and 5-9, but a number of them, especially those nearest the pier, tend to stay open all day and on weekends, particularly during high season. Most of the shops are closed Sunday morning.
Don't pay much attention to written or verbal offers of "20% discounts, today only" or "only for cruise-ship passengers" -- they're nothing but bait to get you inside. Similarly, many of the larger stores advertise "duty-free" wares, but these are of greater interest to Mexicans from the mainland than foreigners since the prices tend to be higher than retail prices in the United States.
A last word of caution: don't buy black coral -- it's an endangered species and really overpriced on the island.