Why not electricity or gas?

When the world on your island is real (no fantasy world), then maybe a smartie (or several) installed an electric station in their house. If there's electricity run street illumination, appliances in the houses, etc. then that's no obstacle. Also the power for the car station can come from photovoltaic plates on the roofs.

And when people cross over to the mainland regularly (causeway, auto ferry) – to go for work or (weekly) grocery shopping, they can tank at a regular gas station then.

If it's fantasy... then I'm out.