Looks like a lot of people are having this issues when I did a google search.
Basically, five years in to owning this GE side-by-side with a chilled water/ice dispenser, the water dispenser stops working. From what I read, most often the culprit is the tube degrades and tends to freeze, thus disrupting water flow.
Band-aid solution I came up with:
1. Fist unclog the iced clog by shooting hot water up the pipe.
To do this you have to disconnect the tube under the door. This is what I used to shoot the water up the pipe:
The water will still freeze after about an hour or so. You can shoot hot water again but the easy thing to do is stick some wire of weed whack line up the dispenser. When you want to get a drink, pull the cord out. This will leave a gap in the ice for the water to flow:
