Possibly. A private nonprofit must pass three tests: it is an established nonprofit (shown by an IRS ruling letter or state documentation in effect on the declaration date), it owned or operated a facility at the time of the incident, and that facility provides an eligible service, either critical or a noncritical but essential social service. Houses of worship can qualify.
An IRS ruling letter in effect on the declaration date granting exemption under section 501(c), (d) or (e); or state documentation that you are a non-revenue producing nonprofit organized under state law; or, if exempt from both, articles of association or bylaws plus a certification of compliance with section 501(c)(3) and state law.
The IRS ruling letter must have been in effect on the declaration date. Organizations that let their status lapse, or that are still waiting on a determination, are a common denial, so confirm your status before you file the RPA.