the cooler and fans are ok.
On the power supply board, two 100uf 200vdc caps are popped on top so they are bad.
What would happen If I put in 600uf 200vdc caps instead of the 100uf?
I have them for free.

Otherwise what about using an ATX PS? Can the fridge thermostat turn on the ATX PS? There are 2 wires you short that turns on ATX PS, would the power good line if unshorted turn it off or is it just a toggle?

I also have several AT power supplies that turn on and off using a manual switch. Would the thermostat handle the current flow?
well I just found that the thermostat is a variable resistor that runs from 16k to 22k

 

best to replace like for like,if its a12volt supply, use an atx psu to see if it still chills.what is the number on the large ic, is it a Tl494?

 

looks like PJ494CD

The peltier seems ok along with the fans. Tested with a 12vdc PS

Thing is I have other 200v caps. I have to buy 100uf 200v caps.
What would happen?

 

Increasing the capacitor value will stress the rectifier diodes and may cause them to fail.
Check all the rectifier diodes before replacing the capacitors.
Timescope.

 

I would also check the rectifier diodes on the right, and test the two transistors on the heatsinks for shorts

 

You also need to replace the small electro next to the big red polyester cap, it is ESR dead too. Possibly the other small electro near it is ESR as well.

Along with the big 200v caps, that will probably get you up and running. You should check the two 150k start resistors near those small electros, lift one leg of each resistor and check their ohms, if more than a few percent high replace them with 1W metal film resistors.