Can an Admin set it so that a guest has to have the Admin's permission to make a account?

Can an Admin set it so that a guest has to have the Admin's permission to make a account? I don't want many people to barge in on our forum topics.

No. you just set it so you have to approve activations,
Or you can have it set for not to accept any new registrations and people can request it by your contact link in your footer. Or just have your site as invite forum only from trusted friends or members of your forum.

Remember you can always register and activate someone yourself,give them a pass to log in with,then once they log in they can change the pass to what ever they want