Allow visitors to create e-mail accounts

I’d like to allow select visitors to create their own e-mail account. Is that possible? I’d want to approve them if possible or require a password to be able to create an e-mail.

Essentially it is a organization and I’d like my members to be able to create their own “vanity” e-mail.

Nope. The admin has to do it. If left up to members (and their friends, I suppose), there’s no telling how many accidental email addresses you’ll end up. If you want to approve the process, you might as well be the one setting it up in the first place.