Would it take a constitutional amendment to change the selection process of Congress from balloting to sortition?
The First Article, Section Four, of the US Constitution says:
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof
That seems clear a state can choose whatever 'manner' it wishes to hold an election.
Of course the law is always open to interpretation. But it seems that constitutionally a lottery would be allowed.
Several states do already mandate a flip of the coin to settle tied local elections.