The short answer is that we don't know. The invention of the bow is prehistoric, so we're dependent on archaeology and rock art. We can reliably say that the bow was already invented by 12,000 years ago (there are two European finds of bows dating back to then: a probably fragment of a bow from near Mannheim, and the Stellmoor bow (from near Hamburg)), but earlier than that is a mystery. There are archaeological finds of small stone points, which might be arrowheads, which would push the invention back to about 20,000 years, over 40,000 years, or over 60,000 years, depending on which find we assume to be arrowheads. The bow and arrow was known and used across much of Asia during the Neolithic. I don't know of any Asian finds of bows of that age, but there are multiple probably Neolithic petroglyphs (drawings cut into rock) showing archery (alas, petroglyphs are difficult to date reliably).

Archery in the Americas appears to result from spread of the technology from Asia, through Alaska and the Arctic. An oldish, but good, overview of the diffusion of the bow in the Americas is given by John Blitz, "Adoption of the Bow in Prehistoric North America", North American Archaeologist 9, 123-145 (1988) (available here).

This late arrival of archery in the Americas (relative to the earlier peopling of the Americas) suggests that the later rather than the very early dates for the invention of archery are correct. While archery was invented by 12,000 years ago, it might not be much older than that.

The summary of the evidence for the invention of archery that I have most recently read was in Christoph Baumer, The History of Central Asia. V1: The Age of the Steppe Warriors, I.B.Tauris, 2012: https://books.google.com/books?id=yglkwD7pKV8C&pg=PA35&lpg=PA35