Hmmmm... Reading up on bows has been very interesting, and a lot of what I read pretty much emphasizes that there are variations in even the same "kind" of bow... as in, a "longbow" couldhave a very different draw weight depending on who you were arming, what tactics it was involved in, and so forth.
So, yeah, "longbows" could be used on horseback... but that longbow would not be the 120+lb draw weight bow used by someone on the ground, and probably have a draw weight of AT MOST 80 lbs.
I think eventually one should be able to define (maybe loosely) the draw weight of the bows you request of the bowyer, and that certain draw weights might require woods (or metals, especially in the case of high-draw-weight-crossbows!) of certain qualities. Or, could use alternate constructions depending on availability of resources or use (like composite bows used by the mongolians, turks, etc.)
For example, if you want a good bow for a horseman, perhaps one could ask the bowyer for a bow of ~50-70 lb. draw weight with as compact a structure as possible (solved by composite construction used by the mongolians, or the crazy looking horseman bows made by the turks). Whereas, if you wanted to outfit a ground unit where they could actually draw high draw-weight-bows and wanted high range for area denial (like the welsh and their longbows), you could order those massive-draw weight ~100lb bows.