The Native Americans had it and the Europeans had it to. Did the Europeans show the natives the weapon? Did the natives invent it before the Europeans came over? Or was it invented before people migrated the land bridge to America
Bow and arrow are definitely pre-neolithic, at least mesolithic and probably paleolithic. We can date arrowheads and sticks looking like bows. Arrowheads can sometimes be disputed to be spearheads and it looks that no arrowheads appeared before 60000BP. The first things interpreted as bows are Magdalenian. That's late paleolithic. Bow and arrow were around at least for 12000 years and therefore already at the time when the first humans entered north america. They are not a "indian" thing.
It is surprising however, that in paleolithicum with anatomicall modern humans (since ~ 200ky) bow and arrow were not there. Spears were. For a very long time hunter/gatherers didn't have bow and arrow.