There are plenty of other groups that do refuse vaccines on the basis of personal religious beliefs. However, it is against the law based on many court decisions to ask you what religion you belong to or upon what you base your non-vaccine beliefs.
This. You don't have to belong to any religious group to have a religious belief. Or you can belong to one that doesn't even condemn vaccines! If you believe that your Creator of any kind has imbued in you a special responsibility to do what's best for your body and the bodies of your children, and you aren't seeing any compelling reasons why some or all vaccines fit this bill, you have a bona fide religious belief.
Also, people often disagree with certain tenets of their religion and still continue to practice that religion. For example, officials from the Catholic Church said that the MMR vaccine was ethically OK despite deriving from the cells of a fetus aborted in the 1960s, but a number of pro-life Catholics (and other Christians) are still concerned and don't get this vaccine.
I actually heard a lawyer argue that even atheists and agnostics have a religious case. He said that if you believe in your gut or the bottom of your heart that vaccinating just doesn't feel right, then you have a religious belief. That's all you need! Some atheists/agnostics may take issue with that notion, but I found it interesting, all the same.
I'm sorry to get so off-topic from the topic from the OP, but I find it infuriating that the WIC official lied to the OP.. And I find it downright frightening that there are continual attempts to undermine Amendment Numero Uno. Freedom of religion means freedom of religion. Unconditionally. It does not mean freedom of religion ONLY if you belong to some recognized organized religion. It does NOT mean freedom of religion ONLY if you sit before some panel of state-sanctioned Thought Police who decide if they approve, (they call this "sincerity testing" in New York State).