As American families plan their holiday meals, officials in Maine would be thankful if they could stop thousands of residents from using food stamps to add soft drinks and candy to their menus.

The New England state is one of several locales that have sought to keep sugary items out of the Supplemental Nutrition Assistance Program, or SNAP, which helps low-income Americans buy food. Nearly 200,000 Mainers, or roughly 15% of state residents, are on food stamps, on par with the share of Americans using the program nationally....