Plumbob,
Having been in the military myself and hunting around a military reservation I've wondered the same thing. Alot of the suggestions you have already received are right on the money. Keep in mind, there is a chance that another military person may have already gotten permission at one time and proved to be a jackass to the landowner. I was just involved with a landowner in this situation. It is up to you to sell yourself with courtesy, respect, and consideration towards the owner and their property. In my case, I not only got permission but was invited back. I made it clear to the gentleman that his rules would be followed to the letter and when I did kill a deer, I cleaned it up and offered as much as he wanted. Not only have a new place to hunt but a new friend that will allow me access to his hunting land, shooting range, and archery range all from doing what we have been taught in the service.