No bad reaction here, I am glad you are asking this question rather than turning them down outright.
You can ask them the obvious questions such as how many years have they deer hunted, how many deer have they taken, etc. But they could easily lie about that. Hopefully they wouldn't since they are deputies. Other than those types of questions, I don't know what else to ask.
As for bow vs. gun hunting, a well placed arrow is just as deadly as a gun shot. I know most hunters don't want to admit it, but it is much easier with modern firearms to take a well placed gun shot than it is by bow. Even with practice, it would be easier to wound a deer with a bow than it is with a gun. I think also, that a less than ideally placed shot with a rifle will cause the deer to die quicker than a less than ideally placed arrow.