If the employer is asking all staff for this information, it is likely permitted. Given that school law regulations permit employers to ask for immunization records during medical exams upon initial employment, it is likely legal for an employer to ask for an update of those records for the COVID-19 vaccine, though of course the regulations did not anticipate these particular circumstances.
However, if your employer is only asking you because you currently have a medical accommodation due to the pandemic, prior to responding you should seek more information from the district as to the purpose of the inquiry as a part of the good faith interactive process required for medical accommodations. You are not required (but you certainly may) to disclose your diagnosis as a part of that process. You may wish to speak with your health care provider about any vaccine questions you may have and if you are deciding to receive the vaccine, whether your medical care provider needs to update your certification supporting your accommodation.
If a staff member such as a school nurse is collecting this information on behalf of the employer, they should be doing so with the authorization of administration or the Board, and not on their own. Any records or responses collected must be kept confidential and separate from a member’s regular personnel file in compliance with N.J.S.A. 6A:32-6.3.