There are a couple of car rental agencies right across the street from the airport.
One option is to rent from one of them before you arrive, walk across the street when you get in at the airport, pickup the car and drive back across the street to the airport and pick up your family and luggage and off you go. At the end of your trip you just repeat the process, drop off the family at the curb at the airport, drive back across the street, drop off the rental and walk back across the street to the airport and away you go.
Another option that I've done more than once is not to rent the car for my whole time there. Instead I arrive, take the airport shuttle to the hotel, settle in for a few days and then once I get the hankering for getting out and about, I go online to expeida or some site and rent the car on Cozumel while I'm already there. Take a cab over to the airport and pick up the car, keep it for the rest of the time I'm on the island, then when it's time to go, drive to the airport, drop off the family/luggage at the curb, drive across the street, drop off the rental and walk back across the street and get on the plane.