Are there natural gas refrigerators?

Yes.

  @Anon: Your price is correct, they start at about $1500, But: How would a Natural Gas refer  work If No Gas Lines? Compressed gas is Propane. Anything  orificed  for methane can't run directly on propane.

 If you have a gas well, yes. This would only be in TX or OK, not TN,. KY, WVA, or AR.

   @Girlie: They get Cold Enough. They also make Natural Gas Freezers. that get to zero F: 

https://bensdiscountsupply.com/natural-gas-refrige...

Natural gas is one type of fuel used in ammonia refrigeration, which is also known as absorption cooling. Other fuel choices include propane, kerosene, butane and electricity. Some manufacturers produce refrigerators that work with more than one fuel choice, such as propane or electricity, to serve customers that may have restricted usage of a single power source. Recreational vehicle owners are one example of this type of customer. Ammonia refrigeration has no moving mechanical parts, and the refrigerator works the same way no matter the fuel source.

When I was a child more than half a century ago, there were such refrigerators. 

I have not seen one for many years.

There are propane refrigerators too.

Yes there are, they are used in remoto places like Alaska and Canada1 Seen them and they work pretty good too!

Yes Absorption Fridges.

They can't cool as well as a compressor fridge though. 

The max temp they will bring it down by is about 30 degrees F

Natural gas and propane/butane as well. They use a small flame to heat a bulb with water and ammonia  that circulates in the system. Weird, but it works. Most campers and motorhomes use both electric and propane heating systems for refrigerators.

Yes, there is a special refrigerator that uses flame and no electricity. I forget the technical word but you can see a big one in the movie Mosquito Coast starring Harrison Ford. The plot is bringing ice to the natives who never saw or felt ice before.

Yep!  Around 1600$US.

Very popular in the Hillbilly states. Where Gas Lines and Electricity grids are not available.

Both natural gas and Propane 

There are indeed, especially used in houses "off the grid" and in camping vehicles.