Kenmore washer ,cold water trickles in, pressure from faucet OK. Must entire valve be replaced?
You probably can not find a rebuilt valve or get the parts to rebuild one. If needed, it will probably need to be completely replaced. There is a good chance you do not have a valve issue. Turn the water valve off and remove the cold water line at the back of the washer. Inside the hose or on the washer itself, there should be a metal screen in the line that somewhat filters the valve and protects it from getting clogged with sediment. Remove the screen and clean or replace it. If the screen wire is not at the washer end try the valve end of the cold water line. While your there you may want to clean the hot water side also. If you are unable to clean the screen, new ones are available for minimal money at hardware stores. Yes the screen wire is necessary, by removing the line additional sediment may very well go into the valve when you turn on the water and cause a valve problem if you don't have one now.
see if you can find a manuel on it also try tracing the wires that go to the valve and back track from there. there is also water level switches and tempture switches that might be messed up