At snack/meal time in my daycare, we have a couple of guidelines:
1. If you want more, you must finish all of what you have been given initially. So no more fries until you have eaten the portion of chicken and veggies first. Then we can have seconds, etc. It doesn't sound like that's the problem, though?
2. We eat at snack/meal time. Not in between. I would not be giving out crackers, etc, personally. For me, it just avoids the "how come HE gets another snack?" discussion. You eat enough at the designated times, or you wait until the next one. If you must have something, you can have all the water you can drink!
3. School age kids are welcome to have something left from their lunch bags. I know that snacktime and lunchtime can be pretty hectic, especially when kids just want to go play. I would double check, just to make sure that lunch kit is coming home empty. If not, there you go, some more to snack on.
4. A snack is just that, a snack. It is not a full meal, nor should it be. Not that you can't have hot, meal-type foods (I think that's a great idea, and will try it out!). But in no way should you be having meal size portions.
If you are concerned, I would mention it to mom/dad, before the child can complain about "starving"!!

Any chance it's boredom? My own kids frequently think they're hungry when they are just bored and looking for something to do.