EEOC to Employees: yes, they can require it.
EEOC guidance makes it clear that employers can mandate the COVID-19 vaccine out of "safety concerns" WITH EXCEPTIONS.
Everything that follows is word-for-word from the EEOC site - note particularly the “Modification” section:
Federal laws protect you against employment discrimination. This Fact Sheet explains how these laws provide rights that can help protect you at work during the COVID-19 pandemic. They are especially important if you are being harassed, if you are “high-risk” and need extra protection from getting sick, if your employer is not allowing you to work, or if you need a modification of your employer’s COVID-19 safety requirements. These laws protect you from retaliation for asserting your (or your coworkers’) rights to be free from discrimination.
I Am Being Harassed
The COVID-19 pandemic has resulted in an increase in some types of harassment. Antidiscrimination laws protect you from being harassed at work because of your national origin; race; color; religion; older age (age 40 and older); sex (including pregnancy, sexual orientation, and gender identity); genetic information; or disability. For example, being harassed at work because you are Asian American can violate the law. If you tell your employer that someone is harassing you at work for any of these reasons, your employer needs to find out if the harassment is occurring and, if so, take steps to stop it.
I Am High Risk and I Need Extra Protection From Getting Sick
If you have a medical condition that makes you “high risk,” or a mental health condition that makes it difficult to come to work, you may be able to work from home as a “reasonable accommodation.” However, to be eligible you must be able to do your regular job from home and your medical condition must be a “disability” under the Americans with Disabilities Act. Many medical conditions can be ADA “disabilities” even if they are not permanent or severe. You might also be able to work from home if you are pregnant, if your employer is letting other people work from home.
If you need an accommodation, you should ask your employer for one. If you can’t do your job from home, you might be able to get a different accommodation, such as protective equipment or scheduling changes.
I Am Not Being Allowed to Work
An employer can’t stop you from working altogether, even during the pandemic, just because you are older, pregnant, have a disability, or you take care of someone with a disability. Your employer can make you stay home if you have COVID-19 and are currently infectious. Your employer can also make you stay home if you have a disability that makes you vulnerable, but only if coming to work would create a significant risk of substantial harm to your health that can’t be reduced through reasonable accommodation. And, even if you can’t come to work because of a disability, you might be able to work from home as a reasonable accommodation. Again, you should talk with your employer if you need a reasonable accommodation. You and your employer may find it helpful to consult the Job Accommodation Network for types of accommodations, at https://askjan.org/
I Need a Modification of My Employer’s COVID-19 Safety Requirements
If you need a modification to your employer’s safety requirements or equipment because of a medical condition or a religious belief, practice, or observance, you might be able to get a reasonable accommodation. For example, you might be able to get a different mask as a reasonable accommodation. Or, if you did not take a COVID-19 vaccine because of your disability or religious belief, practice, or observance, you might be able to get an exception to your employer’s vaccination requirement, and instead ask to use masks, social distancing, schedule changes, or reassignment to stay safe at work.
Find Out More
For more information about your rights, visit the Equal Employment Opportunity Commission (EEOC) website at www.eeoc.gov, or call 1-800-669-4000 (voice), 1-800-669-6820 (TTY), or 1-844-234-5122 (ASL Video Phone).