The president picks the new vice president. The speaker only ascends to the presidency if both the president and vice president are unable to serve. The 25th Amendment and the Presidential Succession Act spells out how it works.
TWENTY-FIFTH AMENDMENT
SECTION 1. In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
SECTION 2. 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 .
Since the Constitution does not indicate an order of succession beyond what is written in the 25th Amendment, Congress passed the Presidential Succession Act in 1792. It was amended in 1947 giving us the following order of succession:
• Speaker of the House
• President pro tempore of the Senate
• Secretary of State
• Secretary of the Treasury
• Secretary of War, 1947
And more on down the line I chose not to list in the interest of briefness.
Anyway, if Harris gets promoted we are even more screwed than we are now. Let's hope Biden can finish his term and be replaced by an non-democrat in the next presidential election.