Easiest thing is, as you say, a stick that you push and pull the door closed with. If you don't want it protruding beyond the edge of the coop (which I can certainly understand, as I know what happens to ME when I'M around setups like that
) perhaps you could hinge it in the middle so that the sticky-out-y part hinges upwards and out of the way when it would otherwise be protruding horizontally. (edited to add: or maybe have the stick not permanently attached to the door, but with a hook or peg at the end that you just reach in and momentarily hook on to a fixture on the door, to give you purchase to slide it open or shut, then remove the stick and store outside the run)
Alternatively what about setting up pulleys, with a loop of rope. Kind of like a clothesline setup? So you would reel the rope in one direction to close the door and in the other direction to open it. Main difficulty would be setting this up so that part of the rope is accessible for you to reel (perhaps you could punt and make a little hatch you'd open to reach in?) and also it might freeze up in ice storms or wet snow.
Or, I forget how high your run is, sorry, but is it possible the door could be slid back and forth by means of a lever operated from above?
I can think of a few other arrangements but not only are they hard to explain without doodling 'em out on a piece of paper, they are not the most efficient or graceful arrangements either.
Whatever rig you end up with for opening and closing it, just make a latch on that 'handle' to hold IT in place, and voila, your door is held in place too. Like, with the firstmentioned 'attach a stick' arrangement, if you just have a hole or screw-eye in the end of the stick so that when you slide the door closed you snap a snap on and the stick can no longer slide.
Good luck, I'm sure you'll figure something out,
Pat