I think sortition is a great idea but you would probably need a constitutional amendment if you wanted sortition in Congress or the Senate. I think the State's have some discretion over how elections are run but I don't think its enough discretion to allow appointment by RNG. I think the strongest argument you could make is its an election where everyone is forced to vote for themselves and tie-breaks are chosen by RNG but I don't think that would be valid because I assume the courts would demand electors execute some agency.

The text of the constitution for electing congress says:

> The House of Representatives shall be composed of Members chosen every second Year by the People of the several States

and there is something similar for the Senate after the 17th amendment. I think pre-17th amendment States may have been able to use Sortion to appoint Senators but it would not have been legally enforceable. The State legislature could pre-commit to elect Senators by Sortition but then they could bail out and just decided to choose who they want when it came to the actual selection.