The best way to search is on Craigslist. It has a checkbox for pets.
If you send me details, I can do a search in MLS. This is the REALTOR database. It also has a filter for places with pets.
I need to know number of bedrooms, price, and location (e.g. Haleiwa to Punalu'u). I don't get a commission or handle rentals, so contact the agents directly from that list.
About a year ago, the Hawaii Humane Society had a list of pet friendly buildings. I guess it's too labor intensive to maintain it, since it's not updated any longer.