U.S. House of Representative vacancies are always filled an election (either a regular election or a special election). See Article II Section 2 of the United States Constitution:
When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies.
Governors can temporarily fill vacancies in the Senate (per the 17th Amendment):
When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
State legislatures may empower their governor to make temporary appointments to Senate vacancies, but not House vacancies.
The House is unique in that it is the only part of our federal government where you MUST be elected by the people. That’s why it’s called “the people’s house”.
We’ve had appointed (not elected) Senators, Vice-Presidents, and even 1 President (*), but we’ve never had an appointed Representative in the U.S. House of Representatives.
(*) Gerald Ford was appointed to the Vice-Presidency under the provisions of the 25th amendment:
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.
And then Gerald Ford rose to the Presidency upon Nixon’s resignation (again under the provisions of the 25th amendment):
1. In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.