Question: I am trying to make an original character who is blind
and hunts with bow and arrow from a flying mount. 
(Obviously, no existing animal can fly while bearing the weight of a human,
so let’s assume that the mount is a mythical/fantastical species;
for instance, a dragon or a gryphon.)

What physical qualities (internal/external)
would enable and enhance the archer’s feasibility and hunting prowess?

Answer: 
Very acute hearing, like that of a barn owl. Barn owls can hunt in complete darkness and without an active sonar like bats.

All owls have one ear pointing up and another pointing down so they can pinpoint sound sources more accurately. Their disc-like face also acts like a parabolic antenna for sound. The ears are hidden beneath the facial feathers, one aimed forward-up and the other forward-down.

Your hunter may wear an owl-mask to help them locate prey, or it may be descended from owls somehow.

Failing that, an active sonar like that of bats and dolphins would be a way to go, too.
Yeah, her/his other senses would need to be heightened, either by natural or unnatural means. Think Daredevil for instance. Other than just hearing, his heightened sense of touch allows him to feel variations in temperature/pressure/humidity, his heightened sense of smell/taste allow him to smell/taste particles in the air. These could potentially be factored into your characters story. 

Like I said, it could be natural through training and adaptation, or unnatural through magic or technology.

This is all assuming the character is human. If the character is another species/race such as one with larger ears or longer fingers it might bode well for his archery capabilities. And if the character is not human, you may consider giving the character a creative new organ. Think birds who can "see" magnetic fields which allow them to traverse the globe easier.