Hmmm, reading this gets me thinking...
Instead of adding the checkboxes to the registration form, I would always prefer making it an extra menu item (type "Affiliate program"), because it takes some explanations for potential affiliates anyway, T&Cs, etc. But maybe that's just me, haha.
However, for what ssbuchan wants, and with chances being high that many users check the box - if they know what exactly it means and are serious about it or not - it appears to me that the following procedure might be appropriate:
1. Ok, add the checkbox to the registration form.
2. In the Joomla User Manager options, set
(a) "New User Account Activation" to "Admin", and
(b) "Notification Mail to Administrators" to "Yes".
ssbuchan's wish would almost exactly come true... with the only difference being that the Admin must not only approve partners but every new user. If that's a deal breaker, then sorry... but maybe someone else here can benefit from it?