[Plugin: Google Analytics for WordPress] Add event tracking to submit button

You would want to do two things:

1. Integrate event tracking into the submission process. Point your developer (or yourself) to this page and they shouldn’t have a problem figuring it out.
https://code.google.com/apis/analytics/docs/tracking/eventTrackerGuide.html

2. Set the event up as a goal.
https://www.google.com/support/analytics/bin/answer.py?answer=55515
This documentation is a bit outdated, as it doesn’t account for counting Events as goals, which is now possible. But it shouldn’t be hard to figure out how to do that by the time you are at the appropriate step, and in the GA Goal setup interface.

Hope this helps!