Your religion is protected by the First Amendment.
The First Amendment doesn't apply here. The relevant part of the First Amendment reads:
Congress shall make no law respecting an establishment of religion or prohibiting the free exercise thereof
Congress didn't pass a law saying people couldn't exercise religion. The First Amendment doesn't apply private institutions. It wouldn't violate the First Amendment, for example, if you decided to say "sorry, no Christians allowed here".
There is however legislation that prevents religious discrimination by private entities: the Human Rights Act of 1964. This prevents discrimination on religion grounds. However, what John Hopkins is doing is not discrimination. They aren't factoring in religion when deciding who gets to be a patient. It would be illegal for them to say that Christians can't be a patient, for example.
Additionally, it's not as though they hate religions apposed to vaccines. They have legitimate reasons to only treat vaccinated patients: more revenue, less workload (it's easier to manage a hospital if you don't have to deal with each individual's vaccine preferences, image (supporting anti-vaxxers might make them look bad), and they might beleive that vaccines reduce the amount of care they need to provide, due to preventing diseases (you may disagree with that, but as long as the hospital beleives that, the point is valid).
Maybe it violates some other law, or moral view, but it's not a religious legal violation.
Their discriminating on what someone doesn't do: vaccinate. The religious argument is why some ppl don't vaccine, is all. Hence, not accepting " those with religious exceptions" is a discrimination to refuse service for not doing something because your religious reasons don't accept it.