hey Quinzy,
I looked around a bit, and I couldn't find anything that asserted firing bows horizontally
The
Codex Mendoza apparently addresses bowmen, but I looked through all of the images I could find and didn't see anything I identified as a bowmen. But like Blue said, these codices are VERY formalized, so it seems unlikely that they would portray the actual use of a bow (since it really wasn't a well respected weapon).
Bowmen are shown firing vertically in the
Lienzo de Tlaxcala from before 1590 -
Not the same as finding it in a codex or a wall painting, but it might be the best available.
Personally, while I know a lot about most modern weapons (like rifles

), I'm not too experienced with bows. But I can't figure out why someone would prefer to fire a bow horizontally. Sure, it's possible, but not very effective.
Aside from advanced composite or compound bows, powerful bows tend to be fairly large - requiring you to draw them pretty far. Firing it underarm severely limits your draw, which limits range, power and accuracy. And firing it overarm presents other problems - if you you're not on your back, your neck gets in the way - and if you fire it with your palm down, the top of your forearm gets in the way.
On the other hand, it does make sense to draw the bow half way horizontally, hold it while waiting for a target, and then extend the arm -rotating the bow vertically (or close to it) - and fire.