With door out fill the track runs with snug wood like 1x2 just where the door was. Cut 2x4 down to exact thickness of frame and build a frame to fit inside the vinyl frame with a couple of cross peices. Slide that in the hole and screw it to the 1x2 fillers. On the outside apply peel and stick weather strip(1/2 x1/4) to the face of the vinyl frame. I would use a sheet of Hardy siding cement board cut big enough to cover the weather strip and high enough that the top is protected by the flashing there. Pre drill the sheet add a little caulk and screw it up and caulk the edges.
Inside, same weather strip on the vinyl and cover with drywall. if you trim both sides of the drywall and add the plastic edge trim you will have very little filling to do. Trim the one side as needed.
I would use fiberglass batt insulation and what ever caulk you think will come off later, after that you should have no damage to the vinyl frame.