Is there a Wordpress plugin that lets users fill in a form to request an account that then has to be cleared/activated by an admin for the user to get access?
Under "Settings > General" - check the box that says "Anyone can register."
After that, the registration page is publicly available at http://www.example.com/wp-login.php?action=register
This feature is built into WordPress, as GavinR mentioned in his answer, however this requires the user to see pieces of the admin interface. To get around this on a site I deployed about a year ago, I used the Theme My Login plugin, which allows the signup, login, and personal preferences for each user to be inside of your site's front-end theme instead of forwarding to the admin. It also adds additional features.
Once you've changed your preferences in
Settings -> General so that the
Anyone can register checkbox is checked, the Theme My Login plugin takes over. It allows admins to prevent new users from logging in until they're approved, and allows you to redirect users to a specific page once they've logged in.