One of former New York Mayor Michael Bloomberg's gun-control groups this week will begin pressuring the nation's largest supermarket operator to dissuade individuals from openly carrying weapons in its stores, following successful campaigns with other national retailers.

The bid to change Kroger Co.'s policy comes after Mr. Bloomberg's groups, including Moms Demand Action for Gun Sense in America, have had mixed results in backing candidates in elections and trying to influence changes in government gun regulations. The groups...