Hi jmikeyh,
The best advice I can give you about getting permission to shoot deer would be, keep doing your vermin control efficiency, cleanly and without causing any disturbance, be professional. Report back to the landowner with what you have shot and if you notice something like pigeons, crows (cattle farms usually have crow problems) or rabbits building up somewhere let them know and offer to deal with it. Likewise if you get asked to deal with something (provided it's legal) get the job done quickly and without any fuss. If you are seeing deer occasionally chuck it in your report if there is a few of them. Once you have a good relationship with the landowner maybe ask if you can take a deer for the pot and offer them a hauch (make sure you make a nice job of butchering it). It's all about building trust and showing that you are professional in what you do. You will be surprised how much difference this can make. A lot of landowners will be far more interested in someone who is a complete package rather than just after deer. One last thing, do not be tempted to embellish numbers as to what you have seen, they work there every day and will very quickly figure out if you are talking rubbish and then it's game over. Good luck, shoot safe and do a good job.
Regards Dave.