No one should allow a business or non-profit to face a situation like this. Only bad managers put something so important into such difficult circumstances. Why go live the day of the election? Are they insane?
Unless the contract (to build this application) had some kind of clause like "This thing must work perfectly or all the money will be refunded" then this is a great contract to get. Let us consider this cynically for a moment. I wish I could land contracts like this one. Can you imagine building an app that the client will only need for 1 day, and you still get paid even if the thing does not work? The chance for fraud is huge. Who would even allow this? The people pitching this must have been some very talented salespeople (perhaps they were convincingly able to argue that they were such huge Republican supporters that if they lost then that would be punishment enough, as it would be so very sad for them to see Obama win?)
I've worked with a lot of startups and clients and the staff always needs training and you never, ever know what all the bugs are until you've gotten some feedback from the staff who are using it for real. To suggest that a software project is done the day you first give it to the staff is idiotic -- that's simply the moment you move to real world testing.
Some developers might suggest that a really good suite of unit tests might provide enough insurance, but that is not true -- I've worked with non-technical staff who tend to confuse bad UI decisions with software bugs, and if a UI decision is bad enough, I think it is fair to classify it as a bug. If some_important_action is crucial to the staff, and none of the staff can find the link to some_important_action, then the UI decision that hid it is effectively as bad as a Fatal Exception, but no amount of unit tests can tell you that the staff finds the UI incomprehensible.