It is my understanding their egg rate will drop if their food drops. Mine love to run around and eat everything in the yard, plus scraps. I would not consider just the yard enough nutrition though. Sure they may survive but they will be less productive and would probably die sooner too. I did stop letting mine eat all night/day long, due to rodent problems, and it forced them to free range a bit more until they got use to the feed schedule. I don't know why but they are eating less now, even when I leave food in the feeder all day. I know my Buff's were really keen on sitting and eating from the feeder all day but I don't see that happening now. I have even stopped feeding them as many cooked eggs/scraps too so who knows what the deal is. I kind of think they got use to eating non stop and I broke the habit and they are foraging more. I am getting less eggs now too, which I don't mind at all as it is colder and darker now. And I do feed them and they often have food left behind for hours and hours so they aren't starved.
You may want to look into homemade feed blends just to see what they need or what you could plant in your yard to help them but just straight grass may be asking for trouble. Would love to know for sure though