Sure. Back in the day, many homes had absorption fridges and there are 50 year old Servel absorption fridges still running today.
All you need is a 12v power supply (the fridge circuit board requires 12v power, probably less than 5A max, and a 120v outlet to plug it in. No LP gas required unless you want to use LP instead of electric (and that requires venting and fire safety concerns is indoors). You might want to make some sort of cabinet for it and insulate around the sides, but that is optional and mostly cosmetic.
Frankly, though, you can probably get a compressor-type fridge cheaper and it will use less electric power besides. You can buy small compressor fridges for $80-$150, medium size (7-10 cu ft) "apartment" fridges for$200-$300, and larger ones from $300 up. And buy them from a variety of sources with no special discounts for those prices.