Can my nonprofit company make me move out for not signing housing agreement.

Although you listed the topic of your inquiry as an ADA issue, I am doing to respond with some basic landlord-tenant information that seems more applicable based on the facts you provided.  It is usually preferable to have a lease agreement in writing, but just because you never signed the housing agreement does not mean they can just make you move out.  In California, you are still a month-to-month tenant with all the rights provided in the Civil Code.  If you are in certain jurisdictions within the state, the nonprofit may need "just cause" to terminate your right to live there.  Go here for a list of jurisdictions current through September 2017.  If not, the nonprofit would still need to give you 60 days' notice before terminating your tenancy.  (Civil Code 1946.1)  Also, California law requires landlords to go through the unlawful detainer (eviction) process before they can force a tenant out.  California does not permit "constructive eviction" efforts like harassment, shutting off utilities, etc.  I would gather together all the documentation you have that shows you asked for clarification of the housing agreement as well as any documentation about your medical condition.  I would need to know more to evaluate the ADA aspect of your case.