Breath might be the largest conduit of scent, especially if you are taking steps to mask you B.O. This has long been believed to be true- Ishi would not eat for a day in preparation for a hunt to help reduce bad breath.
In the past I have eaten apples while hunting. They are "nature's toothbrush"...keep your breath fresh and tast good too lol. I happened to be chewing Trident Spearmint gum the last time I shot a whitetail. He approached from downwind without detecting me. I also used a top-secret odor masking formula that I will not divulge...well since you asked so nicely haha...I bathe without soap, store my hunting clothes in a trash bag stuffed full of pine needles, and wear alpaqua socks.
All joking aside, it surely cannot hurt to chew gum while hunting, and may in fact help you. Helps me pass time sitting still anyway.