Washington state does not have a personal state income tax. Are you instead referring to the Washington state Working Families Tax Credit? If so, see the page linked below. Read the instructions and FAQs on that page, and there is a button to check the refund status.
https://workingfamiliescredit.wa.gov/get-help/my-dor-help-center/apply-online-my-dor/refund
There is also a video on checking the status:
https://workingfamiliescredit.wa.gov/resources/community-partners-toolkit/where-my-working-families-...
