Hey @gkennedy !
I am not aware of any way to do this natively in Shopify, but you can definitely accomplish this using some apps!
Specifically, you could use our Helium Customer Fields app along with the Locksmith app. The Customer Fields app will give the ability to create a customer registration form that requires approval, so the customer account is not created until it has been verified and approved by you or another staff member. The Locksmith app will give you the ability to full lockdown your site to prevent customers from being able to see the site, or certain pages of the site, until after they have been approved.
Here are some resources you can check out with a little more information:
https://help.heliumdev.com/en/articles/4840273-can-i-restrict-access-to-my-store
https://help.heliumdev.com/en/articles/6311748-how-does-account-approval-work