Is there no problem dumping grey water on the ground in campgrounds? My grey water doesn't have food particles or grease. It's only hand soap, or dishwashing soap.
In private campgrounds where lots are purchased, I've seen many owners install a dry well. You dig a hole and bury a 55 gallon drum which is perforated. Fill the drum with gravel and then cap. The gray water then runs into the barrel and leaches out into the surrounding soil.