N.J. Stat. Ann. § 19:59-3

3. Upon compliance with the provisions of this act, P.L.1976, c.23 (C.19:59-1 et seq.), as amended by P.L.2017, c.39: a. any person meeting the qualifications of an overseas voter may register to vote and may apply for and vote by mail-in ballot in any election held in the election district of this State in which the voter was formerly domiciled; b. any person meeting the qualifications of an overseas federal election voter may register to vote and may apply for and vote by mail-in ballot in any federal election held in the election district of this State in which the voter, or in the case of an overseas federal election voter who was born outside of the United States described in subsection f. of section 2 of P.L.1976, c.23 (C.19:59-2), in which the voter's parent, legal guardian, spouse, partner in civil union, or domestic partner was formerly domiciled; and c. any person meeting the qualifications of an overseas State and federal voter as described in subsection l. of section 2 of P.L.1976, c.23 (C.19:59-2) may register to vote and may apply for and vote by mail in a State election described in subsection m. of section 2 of P.L. 1976, c.23 (C.19:59-2) and a federal election described in subsection c. of section 2 of P.L. 1976, c.23 (C.19:59-2) held in the election district of this State in which the voter was formerly domiciled.