Hay gets moldy quickly, unlike straw. I would stay away from hay as a permanent bedding material, although I have used it in a pinch when it will only be in use for a couple of days.
Chickens often will eat hay, and there are many, many threads on here about folks reporting their chickens have gotten impacted crops from hay - some strands are so long, it gets all balled up in their crop & just can't pass through. I've lost two to this, before realizing what was going on, so I no longer use hay.
Straw is better, however, beware that mites find the hollow stalks of straw strands a nice place to live.