Yes, the House votes, but it would be very unlikely for Pelosi to become President, unless — God forbid — both the President and the Vice-President die.
The House votes, but in a very different way, as specified by the Twelfth Amendment. Each state delegation gets a single vote, so it’s up to the House members from each state to agree on a single vote for President and similarly for Vice President. Whoever gets 26 states wins. Fortunately, there are more Republican-majority House delegations than Democrat.
If the House is unable to choose a President, then the Vice-President becomes the President. (I presume this is the newly elected VP, not the VP before the election.)
If no candidate for VP gets a majority of electors, then the Senate chooses the VP from the top two candidates.
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When does the House vote? Because if the election is not clear, not resolved, then then the House election is not resolved.
The 20th Amendment ends the president's and vice president's terms of office on January 20 at noon.
Laws passed by Congress state that “the electors of President and Vice President of each State shall meet and give their votes on the first Monday after the second Wednesday in December next following their appointment at such place in each State as the legislature of such State shall direct.” This year, that day is December 14, 2020.
States must send their electoral votes to Congress by December 23, 2020. Electoral votes are counted in a joint session of the new Congress at 1 p.m. on January 6. If there is no electoral decision, the House selects the next president and the Senate selects the next vice president, by January 20.