I am the product of one such anti-gun person..my Mother! Her Father, my Grandfather, taught her to shoot. Once she even shot the barrel out of her father's best shotgun while hunting ducks - after she poked it into the mud and didn't check the barrel before she shot it. Now that incident alone didn't cause her rift with guns. Her becoming a Mother (of four) did as well as the indoctrination of a majority of youth in their formative ages during WWII.
My Father was a WWII Naval Officer in command of a Mine Sweeper and saw plenty of Pacific rim action. His "authority" in our house was "Bread Winner". He spent A LOT of time at the office on overtime while my stay-home Mom raised us. Dad's influence on me was latent until I too had a family of my own. Dad and kids just didn't mix in well. Some folks were never meant to have kids. My Mother was. My Father, an only child, wasn't. But he nevertheless was a kind and gentle man on a mission to give his family the best that he could offer.
Fast forward to my youth. My Grandfather taught me to shoot a single shot 22 rimfire rifle. Mom and Dad would drop my older brother and me off at my Grandparent's home for the Summer, while my sister was being born, or for regular visits to give my parents a break from parenthood so they could possibly make my sister and younger brother?!?
Granddad's guiding hand and a very hard to cock, old, very heavy, 22 short rimfire rifle, which was difficult to hold on target at 6, 7, and 8 years old, gave my brother and me the first taste of shooting I ever had. Mom wouldn't allow a slingshot, darts, BB gun - forget it! What she didn't know couldn't hurt us.
Mom and Dad in their 70's, and with the alleged Atlanta Olympics bomber supposedly hiding out on the back side of their mountain in NC (in fact he was!), she asked for and received my 7.5" bbl. SA Ruger 45 LC revolver as a "noise maker" so she could "feel safe" if someone came into their home or garage uninvited.
Do not dispair. Your Granddaughter is in your good hands. You will teach her the way and it will be a Defining Moment for her. Be gentle. Don't rush or push for a specific solution. "Plant the seeds." They will germinate in time.