Micronations/Starting your own micronation/Making an Island/Facilities/Food
If you do not live on the island permanently and have a main base on another body of land, it would be more practical to send routine supplies to the island rather than attempt any of the following.
Food Producing Plants
Freshwater Plants If you use edible freshwater plants, such as rice, you're going to have to keep the plants away from the saltwater that your island is probably floating in (unless it's built in fresh water, such as a lake.) One solution is to plant these above the ground, in which case you're also going to have to give them some of the salt free drinking water for them to survive. If the island has no native soil then it will be required to either import it or generate it from raw materials. Hydroponics is another option.
Saltwater Plants Edible marine plants include the seaweed kelp and the red alga carrageenan. These can live and thrive in salty seawater; in fact, most of them can only live in saltwater.
Land Plants Edible land plants, such as potatoes or wheat, are also an option, but this will only work if your island has soil. For most plants (except those few that grow at the seashore,) you will also have to provide protection from the sea and access to fresh water, unless of course your island is built in fresh water.
Fish are the most available source of meat for an island, although means of cooking them would be less readily acquired. Sushi is an alternative, combining farmed seaweed and raw fish. Shellfish (such as crabs, lobsters, oysters, and shrimp) can be gathered in large amounts by use of cages and traps.
Edible insects are another possible source of food, as they do not need much space and can live off of most foods, but you should make sure that they are edible, that there is food for them, and that you can overcome the "Gross!" factor.