Not until they've taken the 'Oath of Citizenship' and become an American citizen. Then and ONLY then can they vote. Generally, but not completely, correct. In some municipalities, resident aliens are allowed to vote in local elections. As far as I know, your statement is true as to federal and state elections. And, of course, the story here deals with such an election.
I knew that and I've posted the laws several times here on FR as I was a green card holder for many years. I was trying not to confuse the issue since this is not a local election.