You can take this 「せる」 as the causative or imperative form, but the context determines which. It is a bit complicated.
According to コトバンク:
「息子を公立学校に行かせてください」 : "Please make your child go to a public school"
Your son might want to go to a private school for some reason. In this case, you can't allow him to go there. So, you might be asking the teacher to persuade him to go to a public school.
In another scenario,
It is asking someone for a permission to do something (If the situation were special, it might be making allowance to do something.).
So, 「息子を公立学校に行かせてください」 means "Please let my son choose to go / allow my son to go to a public school" In this case, the teacher might be stopping your son from going to a public school.