No, the United States does not need a constitutional right to vote as this is already found in the First Ammedmant. The First Ammendmant guarantees the right to free speech, in specific, political speech. Voting is a form of political speech and is thus already protected and guaranteed by the First Ammendmant.
The US already has a constitutional right to vote, guaranteed to non-felons aged 18 and over, without regard to race, religion, or any other characteristic or creed. No other amendments are needed to secure this right. Too many constitutional amendments would only further confuse, not clarify, the right to vote in the US.