Welcome to the forum.
I can remember having allergy shots when I was a kid. Mom gave them at home in those days.
I see lots of kids at the allergist's office getting their shots. They don't seem to mind too much either.
The only way to find out is call the allergist and ask. Allergists treat kids as well as adults. Pediatricians don't give allergy shots. The allergist can also test the child, just like they do adults, to find out exactly what the allergies are and what needs to be put into the serum. Allergists also do a lot of pulmonary testing and treating too, for things like asthma.