PHILADELPHIA—Lily Eskelsen García, the president of the National Education Association, is using her union’s vast campaign machine to wage a personal fight against Republican presidential nominee Donald Trump.

As a Hispanic married to a Mexican citizen, she was insulted by his derogatory comments about Mexican immigrants. As the daughter of a U.S. Army veteran, she was upset by Mr. Trump’s criticism of the parents of a Muslim veteran killed in Afghanistan. And, as a woman, she was offended by Mr. Trump’s comments about how...