JSON API and notification about registration

Set your query var notify to user (if you want to send an email to the user only) or both (if you want to send it to the user and the admin).

Within the plugin controller/User.php:165 calls wp_new_user_notification with $_REQUEST['notify'].

wp_new_user_notification is (lucky you) a pluggable function. If you have your activation logic ready to use, you can overwrite this function and send a modified email to the user. You can overwrite it within a custom mu-plugin (must use plugin).

• Create (if not exists) a mu-plugins folder within wp-content
• Create a pluggable.php file. All files within mu-plugins behave like plugins, but are auto-loaded.
• Create your version of wp_new_user_notification

If you don’t have an activation logic in hand, you can use the answer in the question @Emil described in his comment.