Washington does not have Income Taxes.  You can use the two letter abbreviation WA to complete that entry.

I had the same issue in Alaska and found a solution: When doing the "federal review" (at the end of the process) it will flag as issue each W-2 that has Local taxes and no associated state (skip those!). Then go to the bottom where it will allow you to manually enter info on Associated State for all W-2s (and you can input "AK" or which ever state you need). This resolves the issue and is confirmed to work by TurboTax help specialist.

The system doesn't allow you to select states without State taxes, but somehow there is input in Boxes 18-20 for local taxes.

Boxes 18 -20 are specifically for Local or County taxes.  These could occur in a state that does not have a state income tax.  In that case again you can enter the two digit abbreviation for the state and continue to enter the rest.