[Don’t forget that the International Horn Society has various horn competitions that take place during annual workshops; ditto regional IHS workshops]
Bakitone - worldwide lists of music competitions. Currently says there are 858 competitions in 21 disciplines in 65 countries, but of these only 1 listed for horn
1st International IMMA Records Classical Music Competition (for the best CD recording) - application deadline: Aug. 1, 2015
MTNA Chamber Music Competitions Deadline: Sept. 10, 2014
National Foundation for Advancement of the Arts (NFAA) - supporting young (high school age) artists
List of Music Competitions - by Musicaneo
Classical Music Competitions - list by Musicalchairs
National Federation of Music Clubs - Competitions - They are heavy on the competitions and awards for piano, voice, and strings, but they do have a bit left over for orchestral instruments. See their Competition Chart that lists all awards (pdf file). They also have awards in composition, music education, music therapy, and alleluia, there is one award specifically for "French horn" - see p. 6 of the Competition chart: the Elizabeth Paris Award - $1200 for students 19-26 years old; deadline is April 1 (sheer, coincidence, no doubt).
The International Horn Competition of America
The oldest (founded 1981) most prestigious horn competition in America. It has two divisions: university and professional.
International Independent Music Competition - participants must be under 23 years old.