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.
Some of the state parks we have stayed at have catchment basins for gray water. They usually have a mesh covering over a concrete pipe. We wash our dishes in plastic tubs (usually outside, but inside if the weather is bad) and then dump the water in the designated spot.