Regarding rentals in general, Craigslist has always been my go-to as a first place to look. Padmapper is another great option for listings. Both sites have checkboxes that allow you to search for properties that allow pets.
It might be difficult to find an apartment that allows that many pets. There are plenty of pet-friendly buildings in SF but most of them, in my experience, have 2 pet maximums. With 5 total pets, I'd suggest you focus your hunt for single family homes. Someone renting out a home that they own will likely be more open to negotiation when it comes to pets, as opposed to a landlord with a mulit-unit building, a rental agency, or a landlord with multiple buildings who might have more strict rules.
Sometimes single family homes are listed for rent on the MLS as opposed to Craigslist and other similar websites. If you'd like me to do a quick search and see what's out there in your price range that you might not have easy access to on the internet, let me know and I'd be happy to help!