Are there any reputable chess openings which begin a4, b4, g4, or h4?
These would be classified as A00: Irregular Openings. 

Starting with 1. a4 is Ware's Opening; 1. b4 is Sokolsky's Opening; 1. g4 is Grob's Attack; 1. h4 is the Desprez Opening.

As you've noted, these are not played often. The database at chessgames.com shows only 548 games starting with 1. b4, 260 with 1. g4, and just 10 each for 1. a4 and 1. h4. White has only a small advantage within all A00 openings, 38.5% to 35.1%.