Not yet!
Federal laws PROHIBIT mandating vaccines while they are not yet licensed but only available under an Emergency Use Authorization (EUA). All Covid vaccines fall in this EUA category at the moment. There are no Covid-19 vaccines that are licensed by the FDA, and they are all still in the experimental phase while clinical trials continue. Employers are thus setting themselves up for lawsuits from employees if they are mandating the Covid-19 vaccines at this time. More info:
Can Your Employer Make You Get the Covid-19 Vaccine?
Contact the Informed Consent Action Network
Equal Employment Opportunity Commission Covid-19 Vaccine Mandates Check Section K about religious exemptions
Scroll down in this link for a form letter for employers who try and coerce vaccination
Vaccine Exemptions Information
Information about religious exemptions from post-secondary education (University of Hawaii, military exemptions, form letters, and work exemptions)