Vines should help, but a small tree would allow better airflow. Think mulberry for chickens. The fruit may stain, or you can get white fruited ones, but the berries will feed he chickens for months.
If you want vines, make a separate trellis to leave airflow between the vine and the coop. If you let the vine grow straight up the coop wall, you will trap moisture against the wall, and ruin it. Not to mention the little suckers and tendrils and such. Uncontrolled vines do damage.
Grapes or kiwis might work as well for shade and food. There are fussy grapes, and native american grapes. The muscadine grapes grow fine without spray in humid weather, but taste a little funky.
Virginia Tech did a study of vines for cooling small buildings and said cinquefoil was best because it does the least damage to the structure. I never checked toxicity on that, though.
I always wanted to try hops. It seems like they will allow more sun than a tree until late spring, then because of the established roots, POOF it grows. You have shade when you need it, and then when you want the sun back in early fall, just cut it down.
But be careful to think of any predator that might climb the vine/trellis.