I know you understand this, but let me state it for others:
NEVER ask government to do anything, ever, ever, ever, EVER. They ALWAYS fuck it up.
If your employer is demanding that you get a vax, present them with a written statement for your immediate supervisor and HR manager to sign, that states something along the lines of: as a condition of your continued employment, they are requiring you to get the __________(specific name) vaccine, and it is part of your duties as an employee to get said vaccine.
Watch them squirm when they see that. They will never sign it (liability) and you will go back and forth with them until they either give you an out or fire you. Then, go collect unemployment from them while you get another job or start your own business.
Pretty simple.