You could pull a Stuxnet and create malware that would spread over whatever devices are used to program the voting machines. I believe they typically just use standard USB thumb drives, and are notoriously insecure.

However, the article talks about "election systems" which encompasses much more than just voting machines. Voter registration records are a prime target and those are usually (always?) connected to the internet. You don't need to actually alter any votes to swing an election. Just delete or alter the registrations of some people who you know will tend to vote the other way.