I think you are wrong. If, for instance, a Congressman dies early on in his term, the state DOES select an interim replacement (usually his wife). The mechanism may vary by state (legislator or Governor), and I don't recall the specifics for Louisiana (I've been living out of state for the last 5-6 years), but I am certain that a mechanism for such replacement does exist.
SMGFan is correct. No U.S. House member has ever been appointed to fill a vacancy (it's prohibited in the U.S. Constitution), only a Senate vacancy can be filled (though in some states, a Governor doesn't have the power to appoint). A special election ALWAYS has to be called for a House vacancy. Both Lindy Boggs and Cathy Long (the last 2 widows to serve in LA) had to win special elections before taking their husbands' seats.
From the Constitution, Article I, Section 2,
From the Louisiana Election Code:
§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 his 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.
Acts 1976, No. 697,§ 1, eff. Jan. 1, 1978. Amended by Acts 1977, No. 545,§ 1, eff. Jan. 1, 1978; Acts 1980, No. 792,§ 1, eff. Jan. 1, 1981; Acts 1981, No. 76,§ 1, eff. June 26, 1981; Acts 1997, No. 1420,§ 1, eff. Jan. 1, 1998.