Article I, Section 4:
The times, places and manner of holding elections for senators and representatives shall be prescribed in each state by the legislature thereof, but the Congress may at any time by law make or alter such regulations, except as to the place of choosing senators.
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So the election for senators and representatives can be federalized, with the requirement of photo identification, no early voting, and the issuance of absentee ballots only to those who can document that they cannot appear in person to vote on election day.
Sounds good to me.
You are correct. All within the Constitution.