Possibly, but they wouldn't make a noticeable number. The antivaxxers crowd is bigger and more motivated by the same nonsense. I wonder why they, including the same people who hate injections, never complained about mandatory vaccines for their kids, a practice that went on for decades in many countries?
For example, most European countries had and still have 10 or more obligatory vaccinations for diseases that were finally defeated thanks to the practice, yet I have to meet one parent who refuses to let their kids receive it. I have good memory, and don't recall of anyone screaming when I was taking the anti-smallpox shot as a kid; it's just the same bullshit planted in their heads through social media.
The covid vaccine is newer, therefore not tested enough? Bullshit! Every vaccine has been "young" at some time. There are no records of people protesting back then about treatments against diphteria, meningitis, polyo and others.
There may be some side effects? Again a pile of bullshit. Lots of medicines can have side effects but people don't protest in the streets about that.
Years back I took too many painkillers for my back pains, unfortunately they attacked my guts bacterial flora so I got an intestinal blockage while experiencing a sudden drop in blood red cells. I spent weeks taking tests and restoratives before it was clear I didn't have leukemia, which is present in my family and is what my father died of.
No, it's not fear of the needle, it's pure and simple politicized misinformation through social media and other sources. Those idiots follow what their leaders say without turning on the brain; no critical thinking, or any consideration for the evidence: they're the perfect army any corrupt politician would love to herd.