There is a very important legal distinction between a "service animal" (which is very narrowly defined--it must be trained for a very specific list of tasks to be included under the ADA, as well as being a dog, under those guidelines) and a "therapy animal" or the broader "assistance animal" that applies to housing. So even if a doctor okays a pet as a comfort/therapy animal, if it isn't a service animal it is still allowed to be barred from businesses and the like. I only know about this because of an issue we had at my 2nd job with someone insisting they could bring their ill trained, unhousebroken, stinky matted elderly mini poodle into an indoor playspace for children under 6 under the ADA (including getting in my face about it when I insisted that the dog needed to be back behind the gate after it growled at a baby even before they made it into the play area properly. So I looked it up, called around for confirmation (and printed out all that shit in case I needed it again--which I did, when someone tried to bring in a parrot like 3 weeks later).
I am not sure what the housing authority defines as "assistance animal" (I know it's broader than "service animal" as defined by the ADA but for all I know that might just mean it can be something other than a dog but still require it to be trained) but before you count on not having to pay for a pet deposit or whatever it's probably worth researching first. Just so you know if you can push your rental company hard or not (I'd say it never hurts to ask to not pay the deposit if you can get something from a doctor).
Per the WA State Tenants Union.