To be clear, and this sounds like a small detail but it's actually practical and important, an ineligible person can win an election *even if he can't actually take office*.
Even if #Trump is ineligible to *be* president, he can still run, and we (well, the EC) can still elect him. It's just that he can't be seated, so presumably the elected VP would be seated instead, and the rest of the succession rules would proceed as laid out.
Nothing can stop Trump from winning except the election process deciding to elect someone else.
That's completely separate from eligibility to actually take office, though.
It's not the job of state election officials to undermine the ability of their voters to elect who they want to elect.