The qualitative difference between a needle that jabs deep and a needle that jabs shallowly is actually very small because in both cases there is a needle slipping between skin.
I have a fear of needles; I just got my first flu vaccine this week because now Flu-mist is a thing and I still nearly had a panic attack just being around everyone getting the jab. I also have no tattoos because it involves a needle jabbing me and putting stuff in my body. “This is the case for a lot of people.” Not everyone afraid of needles is your sister. I’m not sorry if “my logic” is different than hers.
The title of your post sounds like a pretty sweeping statement about antivaxxers. Rereading it you didn’t say that it’s the only reason, but you sure didn’t include anything to indicate that there could be more.