You can fill the quiver by holding it in your hand and right-clicking where it will consume any arrows it finds in your inventory up to the 200 arrow limit. You can take out arrows by shift right-clicking with the quiver at hand.
Notes
If you have a quiver and a bomb bag, both non-empty, you can crouch while shooting to fire a bomb arrow! They don't go as far, but they explode on impact!
Using any method to cheat items in, you could give yourself a quiver with more than 200 arrows in it. The quiver uses the damage value to count arrows.
The quiver doesn't render on your character's chest whether you have a chestplate on or not.
Bugs
On-screen GUI does not show number of arrows inside quiver in total, arrows still count.
Shift-clicking on a quiver or bomb bag in the output slot while filling it from the crafting grid can, under some circumstances, cause loss of ammo. (There shouldn't be a reason to shift-click it, since it's unstackable.)
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