It's probably best that you prepare for a long commute ... you're more likely to find something in a suburban/rural location. I had considerable difficulty locating a property for a client with a 65 lb. golden retriever ... wound up in Copperfield at $1350 with a $500 non-refundable pet deposit.
We're good at what we do as Realtors ... but if the property isn't listed in our search resources, we can't be of much help. This is why I recommend that you look at sites such as Craigslist, in the local weekly newspapers, or in one of those "green sheets" found in many convenience stores. In your situation it's best to deal directly with the owners of the homes.
I'm a dog lover myself (have three), and sympathize with your circumstances ... but I don't see any way that I could be of assistance. I wish you and the dogs well.
You would be more successful looking outside of Loop 610.