I am searching for a Klixon M9KA MRA4720-12027 thermo overload. Can you help me locate one please?

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"Thermo Overload?". What is it. Provide more details please.

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On the side of the heat exchanger is a fan Klixon that turns that motor on when the temp exceed 110f on the side of the heater. maybe that thang is going out or your fan motor needs oiling and is cutting out on overload. Next to the fan klixon is usually a limit. Both look the same. The fan klixon will have a marking stamped into the metal such as F-120 the limit will have L-240.

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Does the clicking noise happen every 30 seconds to 1 minute? If so, the klixon thermal current breaker could be cycling due to a current overload in the compressor circuit. A quick test to see if it is the klixon follows:
Open the refrigerator door and observe the interior light while the clicking is occuring. Observe two complete clicking cycles. Does the bulb dim or flicker real low as if someone flipped the switch off and back on real fast 1 time for each clicking noise you hear? If so, your problem is probably related to what I've outlined above regarding the compressor circuit and it's related components.

I found this post by searching on the capacitor number, JSU21X126AQC. My 2001 Kenmore refrigerator (don't have model# here) compressor didn't run, but I heard it hum for a few seconds and then a click. That's the Klixon overload doing its job. I measured resistance of the compressor leads same as original poster. This lead me to determine that the windings were not open or shorted. The wiring to the compressor also fed the condenser fan, and it was running, so power was at least all the way to the compressor.
What I did find was that the 12uF capacitor was only 4uF. My digital volt meter has a capacitor check range. This compressor circuit is a permanent split capacitor type. Without the proper capacitance, the compressor can't start. I replaced it with a 10uF, 220v AC capacitor I had, and now it runs fine. I wonder how many refrigerators are scrapped because of this inexpensive problem?

use a jumper cord and go directly to the compressor to see if it will run. if it don't run, open up the motor box and see if there is a klixon overload inside. this should be normally closed. if it isn't replace it. the klixon is a thermal overload cutout. it usually trips because of high winding temperatures and protects the motor inside. the compressor is a hermetically sealed unit. no, or little freon will cause the motor to overheat.