Q. (cont.) What would an exceptional justification be? What form is needed to apply for 501(c)(3) to be reinstated?
A. This is not an area of expertise of the AP & P2P Network team. We imagine what you need for reinstatement may depend on the reason for the revocation.
The form for reinstatement appears to be the same form used to file for 501(c)(3) status in the first place.
The IRS has information on revocation and reinstatement here:
https://www.irs.gov/charities-non-profits/automatic-revocation-of-exemption (note links to additional information on this page)
If the revocation was due to failure to file 990s, see: https://www.irs.gov/charities-non-profits/charitable-organizations/automatic-revocation-how-to-have-your-tax-exempt-status-retroactively-reinstated.
The form for reinstatement is the application form. Information can be found here: https://www.irs.gov/charities-non-profits/application-for-recognition-of-exemption
For more information on providing “reasonable cause,” see: https://www.irs.gov/pub/irs-drop/n-11-44.pdf
We hope this is helpful, but we recommend that you contact a tax attorney.