From the information that you have given the water valve would be most likely culprit.
The water valve has two seperate outputs. One for the ice maker and the other for the water dispenser. The one for the ice maker seems to fail all the time.
We could test this a couple of ways. The easiest is to go to the water valve (behind the refrigerator where the house's waterline connects to the refrigerator). Bring a bucket and towels.
Shown below as part 5 -
Switch the hoses (you may have to find a way to adapt the hoses because of different sizes) and the wires so that the ones to the dispenser side will be on the ice maker side.
If you can't adapt the hoses, you can find a hose for the ice maker side and put it into the bucket (you still change the wires). You want to see if pushing the dispenser button for water will get the valve on the ice maker side to let water thru it.
Then, you would push the dispenser a few times to see it you got water thru the dispenser (or bucket). Oh, they are not showing you the waterline to the water dispenser in the diagram above.
If you do not get water thru the dispenser button, then you would replace the water valve.
I think your model number is XXXXX (unless you are outside of North America).
Please let me know if you would like more information.