I originally expected to discover that the dispenser switch had failed, so I looked around online to find out how to get at the switch. After watching some of the repair videos here, I was easily able to remove the front panel from the refrigerator.
With the panel off, I discovered that the plastic pegs holding the ice dispenser switch to the door bracket had snapped off after 12 years. The ice dispenser switch still worked, but was completely unsupported; the water switch was only being held on by a single peg (and actually fell off when I performed the repair later). I ordered the replacement bracket and had it in my hands two days later.
It's not difficult to remove the front panel with a taped putty knife, but it is more about finesse than force. Note that even a tiny bit of hard water build-up can make it stick to the door. Once under the panel, I needed a screwdriver to pry off the blade connectors on the light socket, and to pry out the connector for the heater. With all the connections detached, the bracket popped out easily and the new one went right in -- quick and painless.
This was a very easy repair - maybe 5-10 minutes. I actually spent more time cleaning the previously inaccessible crevices I discovered after removing the front panel!