With the passing of Republican congressman-elect Luke Letlow to COVID-19, how will his seat in the House of Representatives be handled? Is it appointed? Is there a snap election?
The governor has the authority to set the date of a special election.
It is worth noting that there will likely be another special House election in Louisiana's 2nd congressional district due to the impending resignation of Cedric Richmond as he was appointed to a position in the Biden administration.
Regarding that election, The Washington Post explains:
Election date: It’s entirely up to Gov. John Bel Edwards, a Democrat. Twelve years ago, the last time Louisiana held an off-year special House election, then-Rep. Bobby Jindal resigned on Jan. 14, a primary was held on March 8, and the seat was filled on May 3, with the victory of Rep. Steve Scalise. If Richmond resigns on Inauguration Day, he’ll probably be replaced by mid-May.
The relevant section of Louisiana law concerning vacant House seats is the Louisiana Election Code, tit. 18 §1279.
§1279. Vacancies; representatives in congress
When a vacancy occurs in the office of representative in congress, the governor shall determine the dates on which the special elections shall be held and the dates of the qualifying period and shall issue a proclamation ordering a special election and specifying the dates on which the primary and general elections will be held and the dates of the qualifying period for the election. Immediately thereafter he shall publish the proclamation in the official journal of each parish in which the election is to be held. Within twenty-four hours after issuing the proclamation, the governor shall send a copy of the proclamation to the secretary of state, who shall within twenty-four hours of receipt of the information notify all election officials having any duty to perform in connection with a special election to fill such vacancy, including the parish boards of election supervisors for the parish or parishes in which the vacancy occurred. The election shall be conducted in the same manner and at the same places and the returns shall be certified as in regular congressional elections. If at a primary or general election in a congressional district one representative in congress is to be elected for a full term and another to fill a vacancy, the ballots containing the names of the candidates shall, as a part of the title of the office, designate the term for which the candidates are respectively nominated.
The US Constitution explicitly allows (but not require) vacancies in the Senate to be temporarily filled by the executive of a state. Article 1, section 3, clause 2 of the Constitution says
... if vacancies (in the Senate) happen by resignation, or otherwise, during the recess of the legislature of any state, the executive thereof may make temporary appointments until the next meeting of the legislature, which shall then fill such vacancies.
Note the use of the word "may" rather than "shall". The constitutions of 37 of the states enable that state's governor to make temporary appointments to fill vacant Senate seats. The other 13 states require a special election to fill vacant Senate seats.
The Constitution explicitly requires some kind of election to fill vacancies in the House of Representatives. Article 1, section 2, clause 4 of the Constitution says
When vacancies happen in the representation from any state, the executive authority thereof shall issue writs of election to fill such vacancies.
Note the use of the word "shall" rather than "may" in this case. Unlike the Senate, which the Founding Fathers did not envision to be an elected body, the Constitution does not allow executive appointments to be made to fill vacancies in the House of Representatives. Also unlike the Senate, the Constitution allows vacancies in the House of Representatives to be filled while the legislature is in session, but only by an election.
One could call such an election a "snap election", but the standard term is a special election. How "snappy" those elections are varies from state to state. However, time must be allotted for candidates to declare their candidacy and then campaign, for election officials to print ballots / program voting machines, and possibly for a primary special election before a general special election.
Louisiana law RS 18:1279 has a specific provision for filling the vacancy of a representative in congress. The governor chooses the dates (qualifying period, primary, and general) and then they follow the rest of the normal election procedures.
When a vacancy occurs in the office of representative in congress, the governor shall determine the dates on which the special elections shall be held and the dates of the qualifying period and shall issue a proclamation ordering a special election and specifying the dates [...] The election shall be conducted in the same manner and at the same places and the returns shall be certified as in regular congressional elections.
A special election. For example see CNN. The date has not yet been set. The office of the Governor will organise the special election.