I advertise my cottage rental on:
www.HomeAway.com
I have the option of allowing pets by indicating 'pets considered' which then appears in the ad. You can refine your search to include pets allowed.
HomeAway has a huge database of rentals across the US. I have even seen local Real Estate Agency's advertising their rentals on HomeAway.
We no longer allow pets due to a bad experience with one of our renters. Allegedly, the next door neighbors dog attacked the renters dog. Although there is a leash law, a lot of people(including our renters) allow their dogs to roam off their properties. You should be able to find something that will meet your needs.
Good Luck.