This is one of those situations where the letter of the law is on her side, but the cost of insisting on it would likely be very high. Time to learn a valuable lesson about picking your battles, and frankly, this time, with this particular decision, it is not worth it.
It's an infectious disease, and a job with customer contact. Vaccination is known to slow the spread and minimize the symptoms, and it is extremely likely that certain classes of employers, including this one, will be allowed to require this particular vaccine once it obtains full approval, which will not be long from now, as it is being so widely used that the data can be gathered exponentially more quickly than is usual.
So, answer is, go get a vaccine ASAP, and provide proof to her manager. The owner is unlikely to try to have her fired on these grounds because his lawyer will tell him it's a bad idea, but it's almost a sure bet that unless he's a known blowhard who always forgets confrontations like these as soon as they end, then he will come back at her with a different set of grounds that is much harder to argue with (and that she does not have an audio of.) MUCH better to get her manager on her side with proof that she's vaccinated. Besides that, in most US states the employee at will doctrine applies, and he probably really doesn't even need a reason to demand that the manager fire her.
It's all well and good to fall back on the FLSA and all of the corollaries to it, but the harsh reality is that unless the employer's behavior is demonstrably criminal, complaints like this are only likely to succeed when you are pretty high up the corporate ladder and have some reputational clout behind you. At the start of your career (or for a part-time student job) it's an insanely bad idea to take something like this up the ladder in a franchise company. As a PP noted, she has now demonstrated unreliability by getting upset and leaving early. She should either quit the job with the required notice if she thinks the place is toxic, or suck it up, get the jab, and chalk it up to a learning experience.
FWIW, my employer, while not yet requiring it, is STRONGLY ENCOURAGING covid vaccination, and asking us to send in copies of our vaccination proof to be added to the personnel file. The message is clear, and only a fool would fight it unless they had a truly valid documented medical reason to refuse.