If that vote is a tie, the state loses its vote. There is no limit to the number of ballots in the House. If the Senate also fails to elect a Vice-President, the Succession Act of 1948 provides that the Speaker of the House shall act as President until a President is chosen. The law was enacted under authority given to the Congress by the Twentieth Amendment. Although the procedures for the way the House and Senate vote are set by the Constitution, there are no rules governing how individual members of Congress vote in such contingent elections in the House and Senate.
Members are free to vote as they please within their state's delegation.
It is conceivable, under these circumstances that the House might select a President of one party and the Senate, a Vice-President of the other. The Electoral College has been the subject of much discussion over the years. It lacks neither supporters nor critics. Opponents of the college call it undemocratic. These critics point out that in a contingent election, the single representative from the least populous state has a vote that carries 54 times more weight than that of a representative from the state of California, the most populous state, with the largest number of representatives.