Are your kids allowed free range of the kitchen? I always keep bananas, water bottles, string cheese, apple sauce, and other healthy snacks within the kids' reach and they are allowed these things whenever they feel they are hungry. Junk snacks like juice, chips, popcorn, and candy are kept on a high shelf and they need to ask for those.
So are your children allowed to take what they want from the kitchen, or do you have strict rules/set times for snacks?
I have noticed a lot of people saying no becasue they don't think their child should be eating all day long.
I do not know how it is in other households, but in mine my kids will eat breakfast then sometime around 10:30 they are a little hungry so they will grab a banana or some grapes. They will eat their lunch just fine and a few hours later want something more before dinner (a couple hours before). They do NOT eat constantly throughout the day. So I am not worried about them eating all of the snacks the second I bring them home from the store. They just nibble when they are feeling a little hungry.
My kids are both pretty small for their ages. They eat pretty healthy and make good choices when it comes to their snacks.