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"In case of the removal of the President from office or of his death or resignation, the Vice President shall become President," reads the U.S. Constitution.

This has happened throughout American history. When President Abraham Lincoln died, his vice president, Andrew Johnson, became president. When President William McKinley died, Theodore Roosevelt took his place.

But if the vice president becomes president, then what happens to the empty vice presidential slot? It’s not surprising people would be confused by this question. For much of American history, the government didn’t have much of an answer. The vice president simply wasn’t replaced because a process wasn’t codified in the Constitution.

"Our Constitution's framers created the vice presidency almost as an afterthought," explains the U.S. Senate website. "The 20th century began without a vice president. Vice President Garret Hobart had died in November 1899, leaving the office vacant, as it had been on 10 previous occasions for periods ranging from a few months to nearly four years. The nation had gotten along just fine. No one much noticed Hobart’s absence."

"The absence of any provision for filling a vice-presidential vacancy had become intolerable in the modern era," continues the U.S. Senate website.

So after President John F. Kennedy was assassinated in 1963, Congress passed the 25th Amendment to the Constitution. According to the new amendment, if a vice presidential vacancy opens up, the president can nominate a replacement.

"Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress," reads the 25th Amendment.

This rule was first put into practice in 1973, when Vice President Spiro Agnew resigned. Richard Nixon was president at the time, and he nominated Gerald R. Ford to take over the role. The House and Senate confirmed the nomination, and Ford took office. When Nixon resigned the next year, Ford became president and nominated Nelson A. Rockefeller as vice president.

So if Biden became president and left office early, Kamala Harris would indeed become the new president.

But Nancy Pelosi, the speaker of the House, wouldn’t become vice president. Instead, Harris would choose the new vice president, and the House and Senate would vote on her choice.

https://www.politifact.com/factchecks/2020/oct/11/facebook-posts/if-president-biden-left-office-nancy-pelosi-would-/?source=patrick.net

Could a former president, who has already served two terms, be named as Vice-President? For example, could President Barack Obama be named as Joe Biden's vice president?

Answer:
No. The 22nd Amendment restricts a candidate from being elected to the presidency for more than two terms. The 12th Amendment dictates that a vice president can not be selected if they are not eligible for president.

This makes sense because they'd be next in line if a president resigns or dies in office.

https://www.wusa9.com/article/news/verify/obama-cant-be-vice-president-for-biden-verify/65-9e4eb588-7725-42e0-b852-69da87a69c63?source=patrick.net