There are many ways that stress might contribute to unhealthy weight gain.
A stressful home tends to be more chaotic. Less structure for meals and snacks to be provided by the parents in a nurturing, dependable way.
Stress and arguing at the table makes family meals unpleasant. No one wants to be at a table with miserable people. (Kids who enjoy family meals have better nutrition, and some studies suggest less weight gain.)
Money worries are a major source of stress. Food insecurity (not having enough food for part or all of the month) is a known risk factor for increased weight. (Studies show that kids who get meals at school, and participate in the WIC program have lower rates of obesity.)
Stress releases hormones (the big one being cortisol) that push the body to store energy and increase weight, and likely contribute to cravings for high fat and sugar content foods.
Stress and anxiety interfere with a child's ability to tune-in to hunger and satiety cues. He may over or under eat.
Many children see parents model overeating in response to stress, or they have been fed in a way that increases their odds of eating more in response to stress. (Kids who are overly controlled or restricted in their eating tend to eat more under stress.)
Stressful homes, often those with money concerns may not have the resources to sign up for after-school sports, or may not live in a neighborhood with safe parks and play areas.
and there are certainly more...
Just another reminder that childhood weight and wellness is complex. Getting soda out of schools, or taxing "junk food" will not solve this problem.