I mean, if it is constitutional, shouldn't their be a set period of time before the election in which voting can occur? As it stands now, couldn't a state just set the early voting at January 1st - Election day. As silly as that sounds, what would stop a state from doing that?
The main thing preventing that right now is the primary process, printing of ballots, naming of candidates that must occur before an election in each of the states. You can't run an election before you have a ballot.
If that could be circumvented through state statute (and it would get complicated), I don't know whether there's a federal statute on point that deals with this issue, probably not.
Then, someone will probably bring a due process/equal protection claim. The argument would have to be creative, since it's not coming up in my brain right now, but it could be made.