Ok, how this works is--->Water flows out of the primary water valve into the water filter. It then flows out of the water filter into the secondary water valve (after the water filter). The secondary water valve has two outlets, one for water and one for the ice maker.
When the dispenser is calling for water (lever pressed) the solenoid for water on the primary water valve and the solenoid for water on the secondary water valve are energized and water flows out of the secondary water valve from the outlet going to the dispenser.
So you will need to pull the refrigerator out from the wall, then remove the plastic water line tubing underneath the primary water inlet valve then get a container to hold water in, and see if water comes out of the primary water inlet valve when you press the water dispenser lever.
Here's the primary water inlet valve for your model:
Water Inlet Valve W10865826
Watch this video it shows you how to remove that plastic water line tubing:
This video provides step-by-step repair instructions for replacing the primary water inlet valve on a KitchenAid 5 Door Refrigerator (Model KRMF706ESS01). Learn more from the experts at Repair Clinic.
www.anrdoezrs.net
Let us know how it goes.
Jake