Some of the Native Americans did have bows upon the Norse arrival, depending on what culture they belonged to. The natives that inhabited Vinland (Newfoundland) at the time of Norse contact were known as the Beothuk. The Beothuk had bow and arrow technology, which they used to hunt prey such as sea birds or caribou. In the sagas of the Norse exploration of Vinland mention the Skraelings (the Norse term for the local native population) as possessing bow, and one individual even offerred to trade his ornate bow for a metal knife belonging to the Norse.
However not all native populations did posess bow and arrow technology. For reasons not entirely known to modern historians, the Dorset culture which inhabited the northern reaches of modern Canada including Baffin Island did not have bow and arrow technology. The Dorset instead chose to use heavier lance and spear weapons to hunt sea mammals on the arctic ice. They never adapted the bow and arrow from either the norse or their native counterparts, and their relative lack of such technology may have played a role in their ultimate extinction due to climactic changes and their conflict with the Thule people. The Thule did have bows, which helped them in combat as they defeated both the Dorset peoples and ultimately helped expel the Norse from the North American continent.