Here are the details:
On November 4, 2008, Starbucks will give voters a free tall brewed coffee at participating U.S. stores.
After voting, voters must go to a Starbucks store, where they’ll be given a tall (12 oz) cup of brewed coffee at no charge (limit one per customer).
This is an extension of Starbucks commitment to community through Starbucks™ Shared Planet™. It lets us immediately support customers who care about the same things we do and who want to make a difference. This idea has also come up a number of times on MyStarbucksIdea.com our on-line forum for sharing customer and partner (employee) ideas.
Apparently all you have to do is tell them you voted. I don't see anything about providing proof of voting.
Just remember to get out there and vote. Don't let predictions by the media keep you away from the polls. Your vote counts.