The short answer to your question is yes.
Children can have hay fever, or more specifically, seasonal allergies.
But there's more to the answer than that. Let me explain. We now know that allergies, asthma and eczema are inherited as a group. This type of allergic tendency is referred to as an atopic tendency. Some children will exhibit one condition, while others may exhibit a few, and still others may exhibit all of these conditions at one time or another.