The fill solenoid should "click" open rather than buzz. Could be solenoid fritzing out. They are easy to replace other than R&R'ing the fridge for access, and should be available at appliance parts stores near you including the internet (do a google search on Appliance Parts, most sites use model # of the appliance for access to a detailed parts list).
You can bench-test the solenoid valve by hooking up 110V to the two spades then blowing thru it; use carefully insulated female spade terminals so you don't accidentally electrocute your lips when blowing. Hot lips are not always the best thing