Hi Christopher. That's a really cute little freezer! As far as I can tell; looking at your one picture of the back, it may have the slightly newer "Whirlpool" variant of the rotary compressor. That variant has the electrical terminals on the side of the compressor whereas the others have the connector on the very top. With a good picture of the compressor area we can probably make a better identification. The terminal box on the side rules out the early Seeger rotary unit, but doesn't confirm anything without seeing more of the compressor.

The thermostat on the back has the same trim plate as the replacement Ranco thermostats. It appears that has been replaced. There is a chance they wired that wrong and it never worked; or that their wiring job was done with such poor craftsmanship that it failed after some use.

I would start by looking at the wiring, closely for obvious defects. Then move on to using the multimeter in continuity mode, to look for open circuits in the system. I would first confirm that the thermostat is in fact turning on and closing the circuit between the two wires on the rear of the thermostat. If that is working, then I would verify the compressor's thermal overload device has not gone open-circuit. After that, test that the compressor motor still has continuity between all 3 of its terminal pins.  

If everything checks out and there is still no action at all; then I would try to power the compressor with a test cord and temporary start-relay; to verify it is functional. 

Thanks for the link to my Coldspot fridge repairs! Appreciated; and hope your freezer doesn't need as much work!

... it's for a reason. Not saying they never recover; but whatever caused it to seize up in the first place will need to be fixed or it will just fail again. 

Lubrication issues and overheating are two primary causes. If that one has no fan, and depends on convection cooling, the only thing you can do about overheating is keeping the coils clean. 

Lubrication is at issue if the freezer has lost its charge and has run constantly with no charge or with air in the system. That will result in a seized compressor after enough hours of running. 

The ones which I have un-seized and seen them continue to operate often fall into two categories.  One is where a line is broken and air has permeated the system, causing oil varnish to form during years of sitting with an open line. Those can often be re-activated with 240V and then the oil changed. I've had a 100% success rate with this. Mainly older Kelvinator, Tecumseh and Frigidaire units.

The others which seem to be able to work after this happens are the GE Monitor Top units with methyl formate gas. They tend to have a lot of seized units. 

It's worth a try to boost it with 240V. You could get lucky. I hope so!  Just be watchful for any signs of cooling which may or may not happen. If it runs but doesn't cool, don't let it run for more than an hour or so.