Here's a good primer on the topic of selling and serving alcohol on the east side of Western Avenue in the 19th Ward.
The Illinois Liquor Control Act of 1934, which followed the repeal of Prohibition, sets forth the law for establishing dry sections by election precinct, and the City of Chicago Board of Election Commissioners Guidelines for Local Option Referenda sets forth the process for establishing (and repealing) dry sections in the city.
This is done on a precinct-by-precinct basis, and only the people who live in the precinct that will be affected are allowed to vote on the issue.
Many people would like to see more restaurants serving alcohol located east of Western, however, the people living in that section have consistently voted "no" to alcohol for almost 90 years.
Frankly, a better way politically to deal with this issue would be, rather than trying to change the law in general in any given precinct, to find a party interested in developing a specific property into a restaurant and to work through a process to allow a liquor license for that venue, as was done with the Barraco's on 95th Street, which took over the building that was the former library, and before that, a funeral parlor. No other property was affected by that decision and the neighbors impacted by the decision approved the sale of alcohol at that location.
BAPA History: The “Wet/Dry” Issue