State of Washington has this page explaining how to hire employees- https://dor.wa.gov/manage-business/grow-business/hiring-employees

The basics are that you have to use your Secure Access Washington login and then go to the business license update section which is associated with the WA Department of Revenue.

Once you are able to login, you have to update your business license. They will ask why and the answer is because you are adding employees.

Complete all of the steps and when you are finished and you pay the small fee, they will send a notice to Employment Security and Labor and Industries. Those agencies will take several weeks to issue you id numbers which you can use to set up the payroll service.