I use “Theme My Login” which is a user management plugin that allows the standard WP screens like Login, Register, Lost Password and User Profile to be integrated into the front-end design of a site seamlessly, so you’re not redirected to the standard WP screens outside the main design of the site.
This also applies to the user profile edit screen, and in this case, I have found added fields for “billing details” since I have WP Invoices Ultimate installed.
This sounds close to what you wanted, but I don’t think its possible to add those fields to the actual registration form, but only to the user profile (edit) form.
My Theme Login provides for many user management features including custom emails, you could add a link in the “your account is active” email to the user profile form instructing them to add their billing details.
