A mom-and-pop grocery chain finds itself at the center of the gun debate for allowing customers to openly carry guns inside.

Some gun control advocates are using social media to ask customers to stop shopping at Caraluzzi's Market locations in southwestern Connecticut. 

"It's a really beloved store, so in no way do we want to boycott it. We just want to be comfortable going there, shopping," says Amanda McCorry, of Moms Demand Action. "I think it sends the wrong message. If you're in the dairy aisle with your children shopping, and someone's open-carrying a gun, it kind of puts you on guard."

Not all customers want a change, however. 

"I would be thankful if, God forbid, something evil and demonic should happen and there was someone there, armed, to protect innocent people," says Stephen Healy, of Redding.

State law allows permit-holders to openly carry a gun, but stores have the right to ban them. In addition to Caraluzzi's, Stew Leonard's also allows guns with permits.

While Caraluzzi's management did not return several calls for comment, Moms Demand Action tells News 12 Connecticut that the store has been willing to listen to its concerns.

Caraluzzi's Market locations include Newtown, where the Sandy Hook shootings took place.