There isn't a way to vet that software by using it, but there are other problems which are easily solved with software freedom (the freedom to run, inspect, share, and modify published computer software for which complete source code and build instructions are a prerequisite): the ballot layout could require changes that were unanticipated by the program's developer. A new ballot layout could require changes in every computer that deals with the ballot in any capacity (producing the page description for printing, scanning ballots for voter verification and over/under counts for example).

We can't predict all of the problems that come up and that alone tells us we need to be in the best position to anticipate changes, hence respecting software freedom for all computer owners. Therefore not being able to inspect that the running voting machine is running trusted software is no reason to deny any computer own their software freedom. That's no justification for pushing voting districts into buying new voting equipment which ostensibly supports the needed change.

So in the end the solution remains the same: anyone who cares about software freedom (as we all should) would care that their voting machines run on free software.