Unfortunately, it is not only Oak Lawn. In many areas, the majority of rentals do not accept pets. Condos are even more because of the rules of the building. Yes, they are aware, but it is the condo board and rules that would need to be changed. Sometimes that can happen, but not with rentals. And even condos that allow pets do usually have size and count restrictions.
If you find a building that is pet friendly and not the landlord, there are sometimes ways to offer additional rent for the pet or an additional deposit to get a yes.
It is difficult though. Rental is very competitive. Pets are considered a liability for many landlords, so you need to tip the scales in your favor. If you are better risk than other potential renters, that helps.