Sure, it shouldn't be a problem. You can either use a basic SharePoint Designer workflow or maybe just add another Email Submit data connection.
Whichever way you do it, you basically need to trigger it ONLY when the form is complete. To do this, simply make sure you have some sort of field (hidden or not) in the form template where you track the status of the form. I typically control this status through my submit buttons. I make sure that only certain buttons show at certain times during the workflow, and each one has a different rule set. Each button click typically sets the workflow status to another state (unless it's a simple data update and not an approval). Anyway, at whatever point your status reaches "Complete," then on that final button click, have a rule that submits to the email connection you've set for the original user. The key is getting that user's email address, which you can do by consuming the UserProfileService and getting the WorkEmail node, but that's a SharePoint 2007 feature (MOSS only). I am not sure if InfoPath 2003 is a limitation there. Regardless, if you have a way to get the originator's email, then you can leverage it in your final submit.