Motorcyles and mopeds are exempted from Washington mandatory vehicle insurance law, so no, there is no law requiring you to have coverage.
But here's why you may very well want to get it anyway: You are still liable from the state's financial responsibility law. Meaning that if you hit something/someone and it's your fault, it's your responsibility to pay for the damage. And since even a minor scrape or ding to another vehicle can cost hundreds of dollars to fix, state officials believe that many motorcycle owners in Washington voluntarily choose to carry coverage.
Also, if you financed your bike with a loan, your lender will almost certainly require coverage so the lender doesn't lose its money if the bike is wrecked/stolen/vandalized/etc.