Shipping things to Mexico is not like shipping things within the US or even to most other countries.
- It will be VERY expensive...probably more than the cost of an inexpensive bike
- EVERYTHING shipped into Mexico will have to go through customs and duty will be assessed at whatever random value the customs agent feels like assessing. The more expensive and newer the item, the more duties you will pay.
- If you are not a resident of Mexico, you would be required to have a customs broker to claim your package at customs on this end, when/IF it even arrives in Cozumel - this too will most likely cost more than the bikes themselves and would require coordination and communication ahead of time.
You'd be much better off bringing your bike on the plane as extra luggage (check with airlines on costs & restrictions) or buying one here. Bringing it on the plane may still cause you to pay duties on it when arriving here. I'm not sure if one bicycle per person is an exempt item or not, you'd have to check on that with Mexican customs as well. See what a pain itw would be to try to bring one for a short stay?
There are several bike shops here and you can get a very decent general-purpose bike here for less than $100.
Start with the bike shop on 30th, across from the Pemex gas station.