I am renting out my house in WA state. The lease states that additional roommates are not permitted, but I have reason to believe that an additional person has moved in without the tenant notifying me. Is this legal? What recourse do I have?

The tenant - if you can prove what you believe to be true - has broken the lease. You have grounds to evict. There is also no longer a lease so you can raise the rent or make any other conditions you see fit.

If it is stated in the lease that no additional persons are allowed to move in then you can evict your current renter and move someone else in.. If you don't want to go through that then make the new " roommate" pay a deposit equal to the current renters and also have them sign a lease also.