In a vast Amazon warehouse in central Ohio, the stress might seem to be all about the work. It’s the holidays, and like every other Amazon fulfillment center, the one in West Jefferson is under the gun. Hundreds of employees are working 12-hour shifts, racing to sort and pack thousands upon thousands of items destined to land in living rooms before for Christmas and Hanukkah.

Yet, for many of the workers, there is a deep, unspoken unease. It has nothing to do with the holidays or Amazon. It has everything to do with being immigrants from Haiti in a nation that just elected Donald J. Trump as president.

Many of them live a half-hour away in Springfield, the city that found itself dragged into the presidential campaign after Mr. Trump and his allies spread a debunked rumor that Haitians there were abducting and eating cats and dogs.

It unleashed a raw, painful time for the city. There were bomb threats against schools and hospitals and marches by white supremacists. And thousands of Haitians, who in recent years had fled a violent, impoverished country and settled in the town of 60,000, found themselves wondering if they should continue to live in Springfield.

Now, many of them fear they may not be able to stay in the United States. Mr. Trump has vowed to carry out mass deportations and to curtail programs, such as Temporary Protected Status, that have allowed many of the Haitians to remain in the country at least in the short term. And he has promised to target the thousands of Haitians living and working in and around Springfield.

“Sometimes I can’t sleep, I am so worried,” said Frantzdy Jerome, 33, a Haitian immigrant, who works the overnight shift at the Amazon warehouse.