An electronic voting machine doesn't have to be Turing-complete, surely? It would require especially-fabricated non-Turing-complete chips / ICBs, and some very simple interface like an LED bank and good old-fashioned physical buttons (as a touch-screen LCD unit alone is probably turing-complete), but I'm pretty sure it'd be possible to devise a specialised electronic voting system that couldn't have its behaviour altered without physical modification.
[edit PS] Of course, this is purely academic. You might as well fantasise about a voting system made of Babbage-esque clockwork.
[edit PS] Of course, this is purely academic. You might as well fantasise about a voting system made of Babbage-esque clockwork.