Do not voluntarily resign or agree to be fired, make your employer fire you. Do not sign anything or agree to anything that says otherwise. If you are fired, you may retain more rights, and it is possible you may qualify for unemployment benefits. Your employer should bear the burden associated with their choices.
You have the right to file a religious or medical exemption. You do not need to give your religion nor can your employer ask. Please convey your own thoughts as to why you hold a religious belief that goes against this vaccination. If you are denied do not give up, file again. FOR MILITARY MEMBERS:
You are entitled to request a Religious Accommodation (RA).
A Religious Accommodation requires you to show that your sincerely held "conscience, moral principles, or religious beliefs" have been substantially burdened. It's not limited to any institutional religion, sacred text, or the doctrinal views of any religious leader.
When you make this request, you may be interviewed by your commander, medical personnel, the unit chaplain, judge advocates or others seeking to review the religious nature of your request and your sincerity. These interviews are not confidential. Be honest and respectful.
If your request is denied, respectfully ask who the appeal authority is, and how long you have to make an appeal. Respectfully inform your commander that you intend to appeal the denial.
Keep the process going. Exercise your full rights. If a non-judicial punishment, court martial, or administrative separation action results, you will be provided military counsel or you may retain your own counsel, either alone or in addition to military counsel. There may be pro-bono religious freedom law firms interested in representing you for free. Ensure you are advised on sanctuary and the impact of discharge characterization on benefits.