The problem here is finding a suitable hook on which to fire the function to send the email. If the user role changes when this form is used, that's no problem: there's a WordPress hook for that (so we don't even need a hook in s2Member).
If the role doesn't change, though, there isn't a WordPress hook that we can use, and I recall now that @JediShark couldn't find one in s2Member. (And if she couldn't, it probably doesn't exist.)
So what you'd have to do is include in your form a way of adding something that's otherwise essentially useless, and then check for that. You could add a ccap, for example. But, in this case, I think it would be much better to add some metadata. Then you'd check for when the metadata is added or updated, and use that check to fire the function to send the email.
Of course, you'll also need to have a way of changing the metadata on each billing modification.