OK. My tenant says it is a Kenmore Coldspot, Model 106.57362700 and serial number HRW2640386. That is a mightly long part number -- I hope it is what you need.
Reminder: The frig has been emptied and shut off since last night. It still does not work as of 10 minutes ago. I don't know if the valve is electric or not, but we tried it with the frig plugged in also. Still no water. The icemaker seems to work and the ice auger does turn.
How does the front come off so that I can see inside where the valves and stuff are?
Will I need to get into the front area at all? If so, I will need to remove the plastic frame around the dispenser unit and don't want to break it. How does it latch shut? There appears to be a couple of buttons that you press up inside, but it still doesn't come loose.
If everything is accessible from the rear to fix this, then you may disregard this question.