If we can discuss constitutional amendments to ban gay marriage, then we can certainly discuss amendments to tidy up issues with voting. Also, depending on how you do it, you can certainly have Federal voting laws that don't interfere with the Constitution, including the landmark Voting Rights Act of 1965 or even the more recent Help America Vote Act.

If you wanted to abolish the Electoral College and adopt a nationwide preference voting system, then you'd clearly require a constitutional amendment. If you want to mandate that all state election officials be non-partisan, that all voting equipment, training, and procedures are well-specified and rigorous, that would be well within the scope of an act of Congress. The only question is how you could actually define "non-partisan".