So far, about 10,000 people have applied, according to the company. Applications will be accepted on a rolling basis through March 31, 2015.
Writers have long been drawn to trains, said Edward C. Goodman, author of "Writing the Rails: Train Adventures By the World's Best-Loved Writers," an anthology of train travel writings from Charles Dickens to contemporary writers such as Bill Bryson. "I think it's the combination of the romance and the adventure." Besides, he added, "what can you write on a plane?"
(Read more: AARP targets boomers with travel website)
In "The Tao of Travel," Paul Theroux wrote: "No mode of transportation inspires more detailed observation than the railway train."
The idea for the residency began in December. "I'd love to think we dreamed it up in our last brainstorming session, but it came to us from the literary community," Quinn said, set in motion when the novelist Alexander Chee discussed his favorite place to write in an interview published in the magazine PEN America: "I still like a train best for this kind of thing. I wish Amtrak had residencies for writers."
Jessica Gross, a New York based writer, wrote on Twitter that she shared those sentiments.
Quinn, whose team noticed her tweet and others, responded by indicating that Amtrak would consider a residency program, but would need to test run it first. Gross was offered that opportunity, and in January traveled roundtrip from New York to Chicago on a train she took as a child. The account of her journey, "Writing the Lake Shore Limited," was published in The Paris Review in February.
What followed was a flurry of media coverage and an overwhelming response on Twitter, with more than 20,000 mentions, Quinn said. Soon afterward, the program was formalized. (Chee is scheduled for a residency in May.)
(Read more: Road trippers, jetsetters traveling more in 2014)
Jennie K. Brown, a 29-year-old high school English teacher from the Harrisburg area, is one of the many hopefuls. "I have a newborn baby at home," she said, and between teaching and caring for her child in recent months, little time is left for creative pursuits. "This residency represents an opportunity to concentrate on my craft, to be really focused."
If chosen, Brown said she would work on two manuscripts: revising a novel, "Poppy Mayberry the Monday," for middle-school age children, and continuing to write a Shakespeare curse-inspired young adult novel. Brown has traveled by train before, including a cross-country trip on Amtrak, and said she hopes to take a new route. "I am confident that the hum of the train traveling on the tracks will be relaxing, and that the never before seen desert, mountains and various landmarks will serve as an inspiration."
The program has resonated with the writing community, but the response has not been unanimously positive.
Amanda Castleman, a travel writer and instructor for Writers.com, is among those who have been publicly critical of Amtrak, calling the required granting of rights to application materials too broad. "The romantic appeal," she said, has already railroaded thousands of applicants "into signing away their rights."
The program feels more like a contest to promote Amtrak, Castleman said, rather than "the pure support for the arts" typically associated with residencies. "This contract hits the most vulnerable part of our community—emerging writers—the hardest. I want to live in a world where residencies and contests alike encourage new talent, not strip-mine rights from hopefuls just starting out," she said. "Writers often complain that society devalues our work. That won't change until more of us say 'no' and push back on poor deals."
Quinn said application materials may be used for advertising and marketing, but noted that applicants have full control over what they choose to submit and the request for rights will not pertain to what is produced as a result of the residency. "We're not in the business of publishing," she said.
Sharr Prohaska, a clinical associate professor at the Preston Robert Tisch Center for Hospitality, Tourism and Sports Management at New York University, said the benefits of the program may have a broad reach.
"I applaud Amtrak for this new initiative with the arts community," Prohaska said. The residency not only offers publicity to Amtrak and an excellent opportunity to gain exposure to writers for their professional work, "it will also raise awareness of the many beautiful places that people can travel in America."
(Read more: Airports ramping up downtown rail connections)
—By Tanya Mohn, NBC News contributor
Follow CNBC's Road Warrior on Twitter at