It has nothing to do with needing to files taxes in your own state, that is not related to getting a refund from Washington state. You just need an interpreter to understand why they mention Jan, 1st.
The Jan 1, 2020 date is all about the new fiscal year, you can only file one time during each fiscal year for a refund. So to file for the refund for 2019 this very first time they want you to wait until Jan 1, 2020 before you request a refund for taxes paid between July 1, 2019 and Dec. 31, 2019.
Refund requests:
Qualified nonresidents can request a refund directly from the Department of Revenue for the state portion of the retail sales tax (6.5% of the sales price) they paid on purchases of tangible goods, digital goods, and digital codes.
Nonresidents can request refunds beginning Jan. 1, 2020 for purchases made between July 1, 2019 and Dec. 31, 2019.
The refund request must include copies of receipts and proof of qualified nonresidency.
Requests are limited to one per person per calendar year.
The request must total $25 or more in state sales tax paid. Requests for amounts less than $25 will be denied.
More information about the refund request process is available on our website: dor.wa.gov/nonresidents.
Looks to me like they want to make some interest money by hanging onto that sales tax for a while and of course also reduce book keeping for the state and for the merchants too.