Okay, please don’t send me anymore emails making this point (I don’t mean to be rude, it’s just that constutional questions elicit scads of redundant emails):
Article IV, Section 4 of the Constitution requires that “[t]he United States shall guarantee to every state in this union a republican form of government. . . .” http://www.law.cornell.edu/constitution/constitution.articleiv.html
Your correspondent is right that the word “vote” does not appear in the constitution, but you can’t have a a “republican form of government” without some kind of voting.
This point comes up a bunch too:
The SCOTUS cases dealing with voting rights are “fundamental rights–equal protection” cases, not substantive due process. Basically, this means that if a state chooses to allow citizens the privilege of voting, it must offer that privilege on an equal basis–can’t discriminate against suspect classes without compelling government interest and narrowly tailored means, etc. A state could, ostensibly, deny the voting privilege to all citizens equally. Which would be stupid.