First off, the bow and arrow was in use much much earlier than the middle age. The oldest examples of bows date back 11,000 years and have possibly been in use for 64,000 years.

The Native American migrations happened in multiple waves dating back 40,000 - 16,500 years ago. Some of those migrations likely would have brought the bow technology along to that continent.

http://en.m.wikipedia.org/wiki/Bow_and_arrow#History http://en.m.wikipedia.org/wiki/Indigenous_peoples_of_the_Americas

I've asked /r/AskAthropology about the current views about the earliest settlements in the Americas. All the mainstream articles I come across still are stuck in the narrative of "this theory blows Clovis First out of the water," so I'd like to hear what anthropologists are currently thinking.