Reply To: Limit form submission to once per user

Yes you could do this by using some of the hooks, to check the author of previously entered forms and maybe a redirect if one has already submitted a form. You then would need to set the user role caps for the custom post type so the user can only edit own. You could create a filter on the archive query that only retrieves post from that user too.

As far as GF integration, its my understanding that this is the point of frontend submission and with the rest of the metabox conditional extensions you could do most of what gravity does. Only thing you may need is a notification system for email notifications.