E.B.
I also wondered, like MilitaryMom6, if it's anything school-related. For some kids the school environment causes anxiety - bullying, lack of privacy in the bathrooms (actual or perceived), feeling rushed (hurrying to eat or get to recess or get in line, leaving very little time to go to the bathroom), etc. Anxiety can manifest itself in a lot of ways.
If I were you, I would keep track of these soiling incidents on a calendar (a private one, just for you and your husband). Just observe, quietly, what days or times these incidents happen. You may see a pattern.
I'd also note, on the days the soiling happens, what foods or drinks your son had. Does he eat or drink anything with artificial sweeteners or anything that is called "sugar free"? In some children, those things (sweetened with sorbitol, erythritol, aspartame, Splenda, sucralose, etc) can cause intestinal upsets. My daughter has been unable to have even a piece of sugar free gum since she was little due to the intestinal pain and other digestive and intestinal side effects. She also cannot have apple juice, which we discovered when she was a toddler, due to the same effects (apples are a natural source of sorbitol). Does your son drink a lot of flavored almond milks or soy milks (they often contain a lot of additives, sweeteners, thickeners, etc)? Does he not drink enough water? Does he eat a variety of healthy real foods, or does he exist on rainbow sprinkle pop-tarts and candy?
Rewards and punishments aren't really appropriate for a physical problem, except when that problem is completely under the child's control. Say your 7 year old burps a lot, and loudly. If it's accidental, after a meal, and he says "excuse me" and attempts to keep it quiet, that's a bodily function that is normal. But if that same kid gets a huge kick out of loudly burping in the most inappropriate places (church, when visiting grandma) just for attention, that's a different story, and some behavior changes should happen. So if your child isn't soiling his underwear on purpose, or to aggravate you, or to get out of a spelling test, then a reward won't work or be appropriate, any more than you could try to earn a reward for not having a gray hair. So I suggest you quietly try to evaluate what's going on in your child's life, and his diet, when these incidents happen. Look for a trigger or pattern.