Super big liability issues there. OSHA has already stated that if a business mandates a vaccine and someone gets sick or worse from it, the business could be on the hook big time from a legal perspective. Any issues arising from the vaccine would be considered "work-related" under the law.
It's going to be very hard for the average non-health related business to get away with mandating vaccinations, IMO. I don't see it being widespread. Small businesses that want to virtue signal need to be very, very careful here. The benefit is probably not greater than the risk.