On a cloudy afternoon this summer, Christine Hallquist, a former utility executive from Vermont, listened closely as Danica Roem, the Virginia state delegate who won national recognition when she became the first transgender person elected to her state’s Legislature, offered tips as the pair canvassed a stark residential neighborhood here. Ms. Hallquist is transgender, too, but Ms. Roem’s advice had nothing to do with gender identity. Try a light, rhythmic knock. Leave a handwritten note with campaign literature if no one is home. Try to earn every vote. “I have so much to learn,” Ms. Hallquist said, duly incorporating Ms....