No. The language in the 12th ammendment is “the House of Representatives shall choose immediately...”
However, I may have been incorrect on how the Seantors vote...I think it is one vote for each Senator. Nevertheless, there is still a tie in this case.
They vote immediately...immediately after counting the electoral votes, as described earlier in the Amendment. And they do that in January after the new Congress is sworn in. The full timeline is here.