No it is not illegal.

>Exemptions to the hours of rule for permit and probationary drivers underage 21: Proof of a need to drive during the prohibited hours for employment and/or religion. For an exemption, the motorist must carry a legible certification to indicate this need from his/her employer or religious official on the official letterhead of the business, organization or religious institution, with the signature of the certifying official and his/ her name, title, address, and phone number. (N.J.A.C. 13:21-8.18)

https://www.state.nj.us/mvc/pdf/license/drivermanual.pdf