How do I clean a refrigerator with no removable shelves?
200 proof alcohol to the rescue! (If you cannot get that locally, use the cheapest vodka with the highest alcohol content you can buy)
Pour a bit into an atomiser for best results (but a good spray bottle will do¹), spray a tiny amount inside the fridge and wipe it off with a kitchen towel after a few seconds.  It'll sterilise your fridge and any leftovers you didn't wipe will evaporate naturally.
Warning:
200 proof alcohol is combustible, so don't:

cook
smoke
burn candles
...

while you're doing this and please: keep the kids away and store any leftover alcohol in a safe place.
You'll go from a dirty fridge to the cleanest fridge in a 50 mile radius.
:-)
Note¹: A spray bottle will do, a squirt gun will not: the alcohol has to be nebulised at least a little...  ;-)