Of course it's coercion. Coercion is part of employment.
Your employer wants you to arrive on time and will fire you if you're late. You're being coerced to get up early.
Your employer thinks you should adhere to a dress code or you'll be fired. You're being coerced to dress a certain way.
Your employer thinks your hair should be kept trimmed or you'll be fired. You're being coerced to get a haircut.
Your employer hires you for a job that will require you to go to countries where there are certain diseases. As part of this job your employer requires you to be vaccinated. You're being coerced to get a vaccination.
I don't see any reason why a private employer can't mandate you take a shot as a condition of employment. It's really their call. You can disagree as to how necessary the condition is but the employer can do what they want.