K.U.
Your son is 7 - if he does find out about the license, he may, by the time he is 12, have forgotten all about it and have other interests.
Responsible hunters and fisherman are much more pro-conservation than you might realize. You might want to attend the event with your son with an open mind. You did not mention if this relative is male or female, or how exactly they are related. You did not say if your son's father is in the picture or not. But if this relative is male, and is a responsible hunter, he may be a positive role model for your son. If on the other hand, you truly don't want your son getting involved, than you should state your case to your relative and not worry about hurting anyone's feelings - you are entitled to your opinions and should be able to stand by what you believe in. But you might want to educate yourself on how hunting can actually benefit the environment and conservation efforts - think about questions of this nature that you can ask if you attend this event with your son.
I also noticed that you wrote you don't eat "too much red meat". But you eat some. You are not vegetarian or vegan. Not saying you should be (I'm not either), but it does make me wonder what exactly your objection is. People that I know that are vegan believe that it is wrong to hurt or exploit animals in any way for human gain (food, clothing, etc.). If you are eating meat, than an animal has died somewhere so that you can have a meal. Many domestic animals that have been bred solely for the purpose of being slaughtered for food live far worse lives than animals that live their lives in the wild and are suddenly killed by hunting.
Also, in many areas, deer overpopulation is a serious problem and many deer end up starving to death from lack of food over the winter. Is it worse for them to starve, or to be killed instantly and humanely by a gun or bow? I also know that many hunters will donate what they don't eat to homeless shelters, so there is that added benefit as well. And any meat coming from an animal that was living in the wild will be as "organic" as it can be!
I would also like to add that I am from Michigan, where the opening day of deer hunting season is practically a state holiday. And I am also a veterinarian, so I don't love animals any less for not being against hunting.