I remember those days well, and the series of polio shots we had to have back then. That was way before disposable syringes. And the after-effects of the shot were painful. Your arm would start aching, and it would be very painful to lift it. It took hours for the effect to go away.
I think you are thinking about the smallpox vaccine. That’s the one that put a small circular scar on the arm.
I remember well. My friend had the needle break off in his arm. I rember him holding it with blood seeping thtough the bandage.
I seem to remember the vaccine put on a cube of sugar which was given to us in School.