How to stop emails from sending to registers after submissions?

Can I make it so, when someone registers, their confirmation doesn't go out automatically but is held until I approve them for attendance?

Setting the notification mail will do that. You will be emailed with the notification of a submission but the user of the form will not receive any email.

Setting-up-Email-Notifications

Then you can edit the submission to approve them of their attendance.

-How-to-Edit-Submissions-Made-on-Your-Forms

Hope you find this information useful.

Please let us know if you need further assistance.