Yup. I've uncovered two references to Granato's as the first pizzeria in Chicago, but with a discrepancy regarding the date.
According to
this website, if you use Proquest to search the archives of the Chicago Tribune for the first mention of "pizza and pizzeria", you get the following quote from 1939:
"The only place in Chicago where you can buy Italian pizza is at a little restaurant on Taylor street (sic) near Halsted. There you can watch Tom Granato, for sixteen years the proprietor of Chicago's only pizzeria , concoct the delicacy and carefully deposit it in his big brick oven, slipping it off long handled shovels of well sandpapered wood onto the hot bricks."
I tried running the same search on Proquest but ran into a technical error.
The Tribune archive also includes
this article from 1992, entitled "Dreams Of Old Family Recipe Inspire A Traditional Pizzeria" by Sue Masarrachia. It says:
"When Arlington Heights resident Dan Alberti opened Granato`s Pizza in Buffalo Grove nearly two years ago, it was the continuation of a family heritage.
The son of immigrants from Naples, Italy, his family had a bakery on Taylor Street, which led to the original Granato`s Pizza, founded in 1917. According to Alberti, his mother and grandfather, Marie Alberti and Camillo Granato, are often credited with the introduction of pizza to Chicago."
So those are two different citations crediting Granato's as the first pizzeria in Chicago, one around 1923-1924 (which is consistent with the sign on the postcard quoted above), and the other in 1917. There may also be a discrepancy regarding the first name of its proprietor, unless Camillo Granato was also known as Tom.