The Washington State Department of Agriculture maintains a page on their website specifically for Apiarists.
On it you will find the laws, general information, and a registration form to register your hives.
You can pay with check or cash. Feel free to leave them a note that you would like online payment option for managing and paying your yearly dues. The employee tasked with admistering this wastes days on processing the the checks and payments, and would rather be working on supporting beekeepers.
The association strongly recommends that you register your hives. For 1 to 5 hives it is a nominal cost of $5 per year.