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The paper
New Jersey accepts a statement written at home. Fill in the blanks, print it, sign it.
New Jersey does not accept a fill-in-the-name form on its own. Under the printed words, add a few sentences of your own about how the shots conflict with your child's religion. Type all required immunizations, or name each one. Sign it by hand, in ink or drawn on a screen, then date it.
The paper. Your own signed and dated letter asking for a religious exemption. There is no state form to fill in. It does not expire.
This is a summary of public law, not legal advice. Requirements change, so always confirm with your state before you file.