In her losing 2014 primary against New York Gov. Andrew Cuomo, Zephyr Teachout, then a little-known law professor, ran as a populist Democrat to the left of Mr. Cuomo.

Two years later, Ms. Teachout, 44 years old, is the front-runner in the Democratic primary for the state’s 19th Congressional District. But now she is the one facing an insurgent rival: Will Yandik, a fourth-generation farmer who hails from the region.

“Zephyr...