In Oak Brook there are 3 listings showing up that allow cats or dogs. These were marked by the agents as allowing them - there might be more that are pet friendly but were not marked. The monthly rental price is $3800 to $12,500 (5 BR 6.1 BA house).
In Oak Brook Terrace there are 2 listings marked as pet friendly. Depending on your needs you may want to check surrounding communities.
Feel free to contact me if you want me to send you the listing information or if you want me to do a customized search, otherwise good luck with your search and welcome to the western suburbs of Chicago!