I'm looking for an extension (maybe free?? but not a must) that allows, during registration process of a new account, to put a special code that will automatically match that account with a VIP membership.
It is not quite clear what you mean with "match that account with a VIP membership". But using the following extension you can add an additional field for the code on the registration form – https://www.itoris.com/magento-login-management.html
The code will be displayed in the customer's account in backend.