Unfortunately, that HF area can command very high rents for homes and section 8 only pays about 40% of the going rate for rentals.
My suggestion is to consider a 3-bedroom apartment or condo. There are several in that school district on Kedzie and 183rd Streets which accept section 8 vouchers. It is not a home, but it is a way to get into that district.
Otherwise, I do know of some other section 8 rentals in the surrounding districts. Please feel free to contact me.