How can I unlock my child's Apple Account without email?

If he is in your your family you can change the password to his account using the following:

Open settings

Tap “Family”

Tap the name of the child

Tap “Apple ID & Password”

Then tap “Change {Name}’s password”

Cliojonno wrote:

I have him as a family member with me as the parent, is that enough??

I don't think so. Try How to use account recovery when you can’t reset your Apple Account password - Apple Support. You may have to abandon that Apple Account and create a new one for him, attached to an email address that you can access, and with recovery questions/answers that you write down somewhere and store safely.

Or ignore me and see the advice below. 🙂

tt2