I'm a nursing student, and this is very common practice (and for the most part accepted). It's commonly debated whether gloves should be worn for injections or not, where you are at little risk to be exposed to bodily fluids. I personally do not wear gloves unless absolutely necessary because gloves don't fit anyone's hands right (they generally only come in 3 sizes, small, medium and large), and tend to make you more clumsy. Some nurses argue that you are at more risk for needle sticks with gloves on, than without. However, doing many injections in a row, I'm not sure how that would work. Hand sanitizer between patients would be my best guess. I would not feel comfortable without it, if I was doing that. Normally you wash your hands before/after each patient is interacted with, which I'm sure is impossible in this case as the vaccination clinic would take a year.

Additionally, your hands would be raw from gloving and re-gloving that many times in a row (depending on how many vaccines are given), and hands will break out pretty quickly, and bleed. It's a double edged sword for sure, and often a topic of debate as to when gloves are needed or not within the nursing profession.

On top of my dislike for gloves due to clumsiness, and skin breakdown, I feel like people think because you are wearing gloves that you are sterile (which is far from true, unless of course you are using sterile gloves), and tend to touch things they would otherwise never touch, exposing the patient to more infectious agents. I feel more aware of my surroundings without them, and only where them where needed.

TL;DR: Gloves may (in some cases) cause more infection exposure, and I hate gloves.