Don't do the moped, ever in any port. Locals tend to drive very fast and accidents can happen very quick, you don't want to spend your vacation in the hospital. Key West is the only place I would consider one. Stick with a jeep/car in the islands, $40-60 per day on average. Besides, by the time you rent a couple of mopeds, the price difference may not be much.
I agree with Rocster... go with the jeep and skip the moped. The major rental companies have facilities right at the terminals. If I remember correctly, two years ago a full day rental from Alamo was $80. We toured the island, staying close to the coast, then went to Chankanaab. On our way to town an unexpected storm hit. The streets immediately flooded, covering the many potholes. We saw at least two mopeds go down... one driven by a 'local'.
I have a 5 inch scar on my right shin thanks to a moped in Cozumel some years back. They were out of jeeps so I did the bike with my wife on the back. The ride was nice but the two falls weren't. Nor the damage bill on the bike. You can take a taxi anywhere you want, and probably cheaper than the jeep.
Ditto advice from me!!!! Just to show you how small a world mopeds are in. Picture Cook Islands. Friend of ours does moped plant... wrecks his leg. Another friend who is EHS (ambulance) witnesses and helps out. They don't know each other at all... diff parts of Canada. We were'nt there of course and put story together yrs. later.
When in Cozumel this past April, we rented a jeep for $50. In addition to the above advice regarding safety (which I agree with).... if you are planning to tour the entire island or even go to the ruins, a jeep would be much faster. Its a long drive all the way around the island. There are several major rental companies near the pier, and also several local companies, the choice is yours. We chose a local company for about half the price of the others and were plenty satisfied. Enjoy your day in Cozumel!
"Pay Less" I believe... they have 2 offices, one in the Baraccuda Hotel which is just to the right from the northern most pier. The other is in another hotel down near the southern most pier (Im sorry, I cant recall the name of the hotel, Park Royale maybe??) But there were a few that I corresponded with before going and this one seemed most reliable with answering emails and providing information. Visit some websites for Cozumel, they will all have Rental Car infomation on them. Then email a few for rates, see what you get. Most were roughly $50-$60 for a one day rental. Watch to see that it DOES include the insurance. Otherwise that will cost an additional $10-$15 for the day. I did ask my insurance agent here at home (Allstate) if my coverage was good for rental in Cozumel, he told me yes BUT if it was included in the price I was willing to pay for the rental, take theirs. That way if I had a situation, I didnt have to get caught up in a big mess with it (basically it would be less of a hassle for ME if I had to make a claim, $10 was worth it). If you can not find these guys, email me and I will track down their website for you and send you the link.