A New Nation Votes is a joint project, sponsored by the American Antiquarian Society and Tufts University’s Digital Collection and Archives, that provides returns from elections held in the United States between 1787 and 1825. You can search the database by keyword, candidate, office, year, and state. According to this post at AHA Today, the database is a work in progress. At the moment, it contains digitized data from 18,146 elections (54% of the entire collection).
I took a few minutes this morning to try it out. A search of elections in New Jersey turned up hundreds of results, including data on a 1795 election for Hunterdon County coroner, an 1800 election for Cumberland County sheriff, an 1801 election for New Jersey Speaker of the House, and the 1804 race for U.S. Senate.
The search process is slow (it took several minutes for the data to load), but if you have the patience, this is a valuable resource for scholars of early national political history. It could also be incorporated into student research.