I’m envisioning one of those small elevated coops with an enclosed run. Yes, leave the grass. They will greatly enjoy eating it down and scratching in it until it is bare. They will even scratch up the roots and eat them. You may need to do something different later but for now, let them enjoy the grass.
To me the biggest thing about the coop and run is drainage. A wet coop/run is dangerous and will stink. If you can at all position it so water drains away from it. Avoid putting it in an area that collects water.
If it does get wet and stay wet or if it starts to stink, let us know and give us some information on how big it is, the drainage in the area, and things like that. Photos can really help. There are different techniques to handle that but which is appropriate for you will depend on your unique situation. Keeping it dry is important enough that I suggest you read this article. Good luck!
Pat’s Big Ol' Mud Page (fixing muddy runs):