The least effect on the property probably means no bare areas or post holes. Putting the coop and run in a bed area would enable you to simply spread mulch and stick in a couple of plants when you move the coop. Since there was no grass to start with, none would have to be replaced. Setting the coop on the ground on a couple of bricks or concrete blocks for the base of the legs would eliminate any post holes. Fencing for the run could be with T-posts that are driven into the ground with a pipe driver. They are easy to remove and leave a small hole which would likely fill in with the next rain.
Just about anything you build will probably be too heavy to move and would need to be demolished to remove it from the site. If you use a panel-style of construction, the panels could be disassembled and moved.
You could build the entire operation on an old trailer frame and simply haul it away in time. All that would be left would be the effect of parking in place for a long time.