[Read more about Republican governors in Democratic states.]

Mr. Scott’s popularity fell, however, especially among conservatives, after he signed gun control measures this year. Still, a poll in July by public media organizations in the state found two-thirds of Vermonters supported the law, and nearly half of Democrats had a favorable opinion of Mr. Scott. Only 18 percent of Democratic respondents in the same poll said they had a favorable opinion of Ms. Hallquist, and 55 percent did not yet know who she was.

That may change now that Ms. Hallquist is the nominee, and she is likely to draw national attention — and fund-raising dollars — because of the historic potential of her candidacy. “She’ll raise more money and her message will get out there more,” said Eric Davis, an emeritus professor at Vermont’s Middlebury College. “Even if she doesn’t get elected governor, the greatest contribution of her campaign could be to raise awareness about the issues transgender people face.”

Before she ran for governor, Ms. Hallquist spent 12 years as the chief executive of the Vermont Electric Cooperative, an in-state power utility that she helped to bring back from near ruin. Her transition from male to female took place in 2015, while she was at the helm of the company, and was the subject of a documentary film made by her son.

As a candidate, she made it part of her stump speech, drawing knowing laughs from her female supporters at a fund-raiser this summer as she talked about what it was like to experience life as a woman for the first time.

“I remember the first time after transitioning, a stranger walking by told me to smile — I’m like, ‘Who the heck are you to tell me to smile?’” Ms. Hallquist said. “What my transition has taught me is just how far we have to go.”