How do I get my license after passing the Washington insurance exam?

Licensing is done through the National Insurance Producers Registry

https://nipr.com/licensing-center/apply-for-a-license

When you apply for a license, you’ll need to provide:

• License number or National Producer Number (if previously licensed)
• Social Security Number (first-time applicants) or Federal Employer Identification Number (FEIN)
• Date of birth (individuals)
• Applicant, licensee, license, residency type
• Electronic payment method: Visa, Mastercard, AMEX, electronic check

Keep the transaction number given as you need that to schedule the fingerprinting