They'll probably eat some, but you're still gonna have weeds pop up. I'd recommend that you'd plant plant starts in this area. Here's what you do:
1) Put the chickens in the area with the hay for four weeks. Add more
carbon material as needed to keep it sanitary and smelling good.
2) At the same time you set this up, start your seeds in a
greenhouse or cold frame or whatever.
3) After four weeks move out the chickens and plant your starts down through the mulch into the soil underneath.
If you aren't starting your own seeds, no big deal, just buy plant starts at the garden store or hardware store.
Ideally, you'll have
enough carbon to absorb the chicken manure so your soil won't be too hot. Be sure to wash your produce thoroughly before you eat it.
Since you're just doing a Chicken
Tractor sized garden it will be rather easy to pull any weeds that come up after you've moved the chickens out and planted your garden.