EEOC Title VII requires employers to accept Medical Exemptions and approve accommodations. Traditionally to get an accommodation, you must have an ADA recognized condition, but like with many things to do with COVID, the establishment is throwing the rules out the window.
The federal Centers for Disease Control has made clear that people who have the following conditions may safely be vaccinated against COVID-19:
The only 'approved condition' the CDC recognizes is a severe allergy to Propylene Glycol.
"The most common medical conditions for which an exemption would be required under the ADA are known allergic reactions to ingredients in a COVID-19 vaccine or immediate reactions to a previous dose of one of the vaccines. According to the CDC, a severe allergic reaction is one that needs to be treated with epinephrine, an EpiPen or with medical care. An immediate allergic reaction is one that occurs within 4 hours of vaccination and includes symptoms such as hives, swelling, wheezing or respiratory distress. For comprehensive information on allergies and the COVID-19 vaccines, see the CDC webpage COVID-19 Vaccines for People with Allergies. "
BUT: Allergies are very rare. Even reaction to the first jab isn't good enough to not get the 2nd. And how would one know they are allergic to ingredients in a shot that haven't been given before?
SO, because your medical information should be allowed to remain private, if you can have a Doctor write a note indicating that you have a medical condition AND that in their opinion you should not get the shot (without disclosing that condition) your employer might accept it, unless it's a hospital. They strictly follow the CDC. (because that's going so well)
For this reason, most people are submitting Religious Exemptions. But you can always submit both, to have a backup.