Nathan, I first want to tell you that even bringing up this question is a good sign that you're interested in doing things the right way. Good for you! Asking permission to hunt private property is the way to go, and will do a lot to bolster your reputation and the reputation of the hobby in general. So thanks for asking.
Seems that your biggest concern is that you feel a little awkward about asking for permission. I think you have some options.
1)You could attend the Sunday fellowship that often follows the services. It's usually a very casual get-together which includes the pastor or minister and several of the churchmembers. You could bring a plate of cookies and introduce yourself and tell them what you'd like to do. Getting a few people on your side to nod or smile or express interest in your hobby is a good way to gain some support for your plan. Honesty is the best policy when approaching these folks.
2)You could type up a brief letter with a SASE and ask that they send you a reply by a certain date. Give them at least 6 weeks because they may have to have a meeting of the elders or laymen before an answer could be given.
3)I don't know how old you are or if you are married, but you could take someone with you when you go to inquire at the church or parsonage. Sometimes having an older person or a female with you makes breaking the ice a little easier.
4)You could telephone the minister or pastor and introduce yourself, explain that you've been trying to decide how to approach the subject because you want to be respectful AND that you're feeling a little awkward because you're not a member of the church. Again, honesty is a great way to get your foot in the door. Nothing disarms a person of the clergy like heartfelt honesty.
Please let us all know which tactic you use and how it went. Good luck, and keep the faith. LOL