Yes and no
As much as I know you do work when any body displaces having an act of force over it.
So when you are pulling the bow string and the arrow you are moving the bow string a little with a lot of force applied by you.
But when you release the bow string it again returns to its original place so again no displacement therefore no work done by you on the bow string.
But the arrow moves to some distance with a force therefore Work is done on arrow after releasing the bow string.