You have some challenges. Your best bet would be to work with an apartment complex. Typically private renters put limits on whether animals are allowed and what types. Pets, if allowed, are usually limited to size - 25 lbs or under is the norm - and montly pet rental per animal and pet security deposits assessed at one month's rent are usual.
You can contact renters individually and depending on their personal restrictions, you might be able to achieve your housing goals.