There are a select number of buildings with elevators. You are correct that the older buildings , built between 1890 and 1930 are limited to a max of three floors and are walk ups. there was a building boom in the 60's and 70's those buildings are much taller and have elevators.
There are several buildings in Oak Park with elevators. Most of these were built in the 1960s-70s and may not have / allow in-unit laundry. There are also a few new construction buildings that have garages, elevators, central air & in-unit laundry. 221 N Kenilworth is one of the nicer elevator buildings in a great location close to downtown Oak Park.