I hadn’t known this — it turns out that the New Jersey Constitution of 1776 allowed all heads of households “worth fifty pounds clear estate” to vote in state elections, and therefore in elections for the federal House of Representatives. This didn’t include married women, but it did include widows and adult unmarried women; and women did vote under this constitution, until the franchise was restricted to men in 1807.
There’s a pretty thorough article on this, though available online only to JSTOR subscribers: Judith Apter Klinghoffer & Lois Elkis, “The Petticoat Electors”: Women’s Suffrage in New Jersey, 1776-1807, 12 J. Early Rep 159 (1992). Thanks to Prof. Rob Natelson for the information. Wikipedia also reports that a short-lived Corsican Republic (1755-69) provided for women’s suffrage as well.