I live in WA where there is no income tax, therefore I am unable to enter my employer's state ID number. TurboTax is preventing me from continuing without entering that

Leave all of the state information at the bottom of your W2 blank.  TurboTax is trying to generate a state tax return based on that information and since you entered Washington on there it wants the rest of the information it needs to make that return.  Leaving all of those state boxes empty tells the program that there is nothing to generate.

@samtownsend01