We have lived at Grand Caribe (2 miles north of town) since last August with bicycles only. Sure, it's inconvenient at times, but it can certainly be done -- and for much less cost than renting a golf cart for 6 months. As others have said, the grocery store north of the bridge will deliver for heavy/large items. For everything else, carry in your bike baskets and backpack. We brought a set of two "grocery baskets" with us for my husband's bike that fits on a rear rack and allows us to carry nearly everything we need. We do all our shopping via bikes. These are worth their weight in gold!
Wald Bicycle Rear Baskets 
AmbergrisCaye.com 
