Art. I, Sec. 1 of the Constitution requires that all of the electoral college electors vote on the same day, but does not require that they all be selected on the same day. How electoral college electors are selected is expressly left up to each of the State legislatures; however, the Constitution is silent about when the electors have to be selected by the States. So federal statute spells out when the electors need to be selected.
That is not quite right:
Article 2, Section 1, clause 4 states:
"The Congress may determine the Time of chusing [sic] the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the United States."
So, the Constitution does clearly state that the Congress sets BOTH dates. I guess one could quibble whether the final clause applies to both preceding clauses, or only the last one - but IMHO it is clear that both dates are meant. It would be silly to allow the states to choose the electors "at will", but the electors can only cast their vote on a specific day.
Also, that "may" could easily be abused.
I interpreted the clause "which Day shall be the same throughout the United States" as only modifying the portion of the first clause that reads "the Day on which they shall give their Votes" and not the portion of the first clause that reads "the Time of chusing the Electors."
In other words, the "Time of chusing" does not need to be the same time throughout the United States but the "Day on which they shall give their Votes" does need to be the same throughout the United States.
Grammatically, if the second clause was intended to modify both "the Time of chusing" and "the Day on which they shall give their votes" then the section should have read either, "The Congress may determine the Time of chusing the Electors, and the Day on which they shall give their Votes; which Time and Day shall be the same throughout the United States" or "The Congress may determine the Day of chusing the Electors, and the Day on which they shall give their Votes; which Days shall be the same throughout the United States."
In any event, it is moot because Congress has by statutes set dates for both the appointment of Electors (election day) and the election by the appointed Electors (in December). Congress has also by statute provided for a state to use a subsequent date for appointing Electors if for any reason the state is unable to appoint its directors on election day.