I have a side by side whirlpool fridge with ice-maker and water dispenser; filtered. It's about 8 years old. The water dispenser recently started clicking on and off by itself and running down the front of the fridge. Any ideas of what component I should check, and where to find it.

Look for a short circuit in the wiring or a defective switch that controls the dispenser.

I had been reading a lot about defective water valves. Could that also be it, or would it only cause a drip?

You said the valve is clicking on and off. The electric short is making it click on and off. A drip would ba a bad seat in the valve.

However; couldn't it be a bad celenoid, which looks like it is connected to the valve itself. I would take a look at the switch, but I can't figure out how to remove the front panel of the dispenser

It could be the selonoid also. If the valve is leaking water with no power going to it. Pull the wires off the selonoid and see if the water stops. If it does not stop it is the selonoid valve. If it stops then you will have to find the source of the power going to it all the time.

Now you originally said it was clicking on and off and I gave you the answer to that exact problem which is a short/bad switch etc.

I just bought a similar Whirlpool side by side frige with the same problem. It periodically dispenses about a cup of water each time by itself.