I have a child that is allergic to peanuts, a couple of tree nuts, and shellfish. We do not really know what kind of reaction he would have to coming into contact with a trace amount of nuts because he is 3 and I feel that is too early to do a challenge test to gauge what would happen. In the meantime, we carry an epi-pen with us everywhere (but have not had to use it).
So I do not allow him to have any products that have nuts, may contain nuts, or processed in a facility or equipment that also process nuts (or shellfish). It's rare when we even have such a food product in our house just so we don't have to worry about cross-contamination.
I think age of the child has a role. A 3 year old isn't going to know what foods he can and cannot eat so keeping the child away from foods that may have nuts requires more vigilance. Also 3 year olds are messier so it's easier for a 3 year old to get peanut butter on his/her hands and then touch another kid who might be allergic. As a parent of a young child with food allergies, I check the labels of everything that a parent might bring to class if it's intended to be shared with everyone. I do not check each kids personal snack. If in doubt, I don't let him eat it. I also tend to bring extra snacks that are safe just in case.
An older child (even as young as 5 or 6) may understand their allergies better and have been taught not to eat other people's food. Or for kids not allergic, they can be told not to share their food with the kids with allergies.
In your case, if the snack is meant to be eaten only by your child and it says "may contain" or "processed in a facility...", then I'd say it's probably okay. You can always check with your school if you really want to be certain. If it's likely that it may be shared, then I'd err on the side of caution and not use it for school. Each person's reaction may be different and over time some nut allergies can get worse with each exposure.
There are some online places to get nut-free foods, but I have been able to find plenty of foods in my regular grocery store for my kid. There are a number of cereals that are safe. If you look around, you can find granola itself that is safe, although a lot of the pre-made granola bars are not. There are many crackers as well that are safe. It just takes some effort to look at the packaging to see which ones are safe.
For example, I've seen one brand of pretzels that is not safe, but another in the same store be safe. Same with Veggie Straws. On the flip side, I've seen pre-packaged fresh spinach that says it may contain shellfish.
Typically finding nut-free chocolate candy is difficult as is the pre-made cookie dough. Things that are bread or bread-like (crackers, etc) are often culprits for hidden nuts, but there are a number of items that can be found.
I don't think that companies generally take the easy way out and label their food "just in case". It's more to do with all the products that they make and whether they share the same processing/manufacturing plant. For example, the bite-size hershey chocolate bars are not truly nut-safe, but the regular sized bars are.