New Jersey was the first state to grant women the right to vote. They, as well as men, had to be property holders. Because there was some confusion about what the State Constitution meant on the subject, in 1797 a law was passed confirming that women had the right to vote statewide. See “for whom he or she votes” below.

An Act to regulate the Election of Members of the Legislative-Council and the General Assembly, Sheriffs and Coroners, in this State
Passed by the New Jersey General Assembly at Trenton, February 22, 1797.
[…]
9. And be it enacted, That every voter shall openly, and in full view deliver his or her ballot (which shall be a single written ticket, containing the names of the person or persons for whom he or she votes) to the said judge, or either of the inspectors, who, on receipt thereof, shall, with an audible voice, pronounce the same of such voter, and if no objection is made to the voter, put the ballot immediately into the election box, and the clerk of the election shall thereupon take down the name of such voter in a book or poll list, to be provided for the purpose; and if an adjournment of the poll shall take place during the election, the aperture in the top of the box shall be secured by the bolt aforesaid, and the names on the poll list shall be counted, and the number put down in writing, and the said list locked in the box, and the keys kept separate by two of the persons hereby appointed to conduct the election.

The following poem appeared in 1797 in the “Newark Centinel of Freedom.” It reflects the difference of opinion on the matter of voting rights for women. “Democrats” refers to Jeffersonians. Women did vote in fairly large numbers, but not for long. The Assembly passed a law in 1807 limiting the franchise to white males.

Let Democrats with senseless prate,
maintain the softer Sex, Sir,
Should ne’er with politics of State
their gentle minds perplex Sir;
Such vulgar prejudice we scorn;
their sex is no objection. . . .
While woman’s bound, man can’t be free
nor have a fair election.

See this SOURCE for New Jersey’s actions. Find the poem HERE.