The only constitutional requirement regarding voting is that the ELECTORS vote on the same day. The electors would be the electoral college.
In fact, the constitution does not require, at all, national elections, grant us the right to vote for president, etc.
Article II, Section 1:
Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.
and
The Congress may determine the Time of chusing the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the United States.