The vote for Lawrence doesn’t go through because the terrorist attack interrupts it. He is essentially on a path to become Pope until the moment he attempts to cast his vote. Immediately following that, the cardinals have their emergency meeting in the theatre and Benitez makes his big speech which is part of what convinces a majority of his colleagues to vote for him, the other part being the hole in the window exposing them all to the outside world and reminding them that they are in service to God and his creations rather than themselves. As for why we don’t see Lawrence vote for Benitez, it’s because we’re not necessarily supposed to pick up on the significance of these events until after the plot twist where Benitez is elected. That’s also why they made his reaction to the speech ambiguous and why that election scene is shot in such a way that it initially looks as though Lawrence has won, and when he hasn’t, isn’t immediately clear who has (out of Tedesco and Benitez).