Can children grow out of seasonal allergies?

Sometimes. Some people's allergies get better over years, some get worse, and some are stable. Keeping allergies well-controlled not only keeps kids feeling better and sleeping better, but also doing better in school (it's hard to learn material when you feel miserable)! also, keeping allergies under control decreases the risk of ear infections and sinusitis.
Seasonal Allergies. It depends on what we mean "outgrow" allergies. Children tend to become most allergic between the ages of 10-25, so it is all relative. Some happen earlier, some happen later. Seeing an allergist for testing, or even just treatment would be the best option.