No, there are absolutely propane POWERED refrigerators. Propane goes in, the fridge burns the propane, some goddamn troll physics happen and the inside of the box gets cold. It's called the absorption cycle and they're commonly found installed in RVs.

I've attempted to understand how this works; it has something to do with boiling ammonia out of water and then re-dissolving it. It's a refrigeration cycle with no moving parts that runs on a sufficiently hot source of heat. Some also have electric heaters for when you have abundant electricity to save propane.