I will preface this by saying I'm not licensed in WA, but thought I would try to give you some information since no one from your state has responded.
I'm not sure why or how the landlord would have the ability to demand such information, and evict without it, unless maybe you used your condition as an excuse for late paying or not paying rent? If you are in full compliance with all terms and conditions of your lease, including but not limited to paying rent, I am not sure how the failure to provide this documentation would be a basis for eviction. Now, if you are in a month to month type situation the landlord might be able to terminate your tenancy upon certain notice to you, but that would be different than eviction unless you still refused to leave after the notice period had expired. Best course of action might be to seek a personal consultation with a local attorney who can obtain more facts and provided individualized guidance. You can use the "Find a Lawyer" tab on Avvo to locate someone in your area. Best of luck to you!
I am licensed in the State of Ohio, posting a response to your question or issue does not create an attorney-client relationship and I AM NOT providing you legal advice. You should speak with an attorney who is licensed in your state to whom you have provided all the facts, before you take steps that may impact your legal rights--or rights to recovery of damages.
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