In Quentin Tarantino’s upcoming movie, “The Hateful Eight,” the opening scene called for a stagecoach hurtling across a wintry Wyoming landscape. For the filmmakers, that presented a problem: Where do you buy a working stagecoach?

Turns out one place is just outside of Letcher, S.D., population 171, where Doug Hansen runs a 13-man assembly line, building and repairing horse-pulled vehicles.

Mr. Hansen’s trade might seem to land him somewhere between switchboard operator and milkman. But, in fact, business is booming.

He is on track to build or restore about 100 horse-drawn vehicles this year, working alongside the farm where he was raised. “I was intrigued with the lost art,” Mr. Hansen says.

Delivering a custom stagecoach typically takes Mr. Hanson about a year, he says, and they are priced like a luxury car. The 56-year-old builder is well known among . . . aficionados of the Old West; he enjoys a backlog of orders, he says, that will keep him busy well into 2016.

One of Mr. Hansen’s creations is scheduled to carry VIPs Saturday at the Berkshire Hathaway annual meeting in Omaha, Neb., where fans and investors from around the world will celebrate 50 years of Berkshire under Warren Buffett’s control. The stagecoach—commissioned by Wells Fargo & Co., one of Berkshire’s largest stockholdings—will carry the bank’s chief executive, John Stumpf, and Mr. Buffett’s bridge partner, Sharon Osberg. Mr. Buffett says he relinquished his spot in the four-seater to make room for two of his nieces, who were “very excited” about the offer.

Mr. Hansen is also thrilled. “It’s quite an honor to have something coming from a shop out in the cow pasture in South Dakota,” he says, “and ending up being a Wall Street highlight, so to speak.”

A Touring Concord Coach built by Mr. Hansen once ferried Great Britain’s Prince William and his wife, Kate, the Duchess of Cambridge, he says. Fuji Safari Park in Japan ordered a small-scaled coach that could be pulled by miniature ponies to carry children through a section of the park with a Western theme.

Mr. Hansen’s clients also include billionaire businessman Joe Ricketts, who founded online brokerage TD Ameritrade. Mr. Ricketts asked Mr. Hansen to build him a purple “show wagon,” similar to a traditional stagecoach. Mr. Ricketts sent over a twill button-down shirt in a purple he was sure would make his wagon stand out at the Calgary Stampede, an annual Canadian rodeo.

Mr. Hansen was resistant: A purple-colored vehicle wasn’t historically accurate. Mr. Ricketts asked a fashion-house color coordinator to intercede and aubergine prevailed.

Mr. Hansen experimented with colors for months. Fourteen coats of paint finally yielded a satisfactory hue, close to eggplant. “It was royal,” he says. “It was rich.”

The wagon cost about $50,000 and performed like a champ. The six-horse hitch won the Calgary Stampede in 2010 and 2011.

Mr. Ricketts, whose family owns the Chicago Cubs, says he dreams of asking Mr. Hansen to build him a “cook wagon,” the portable kitchen used by cowboys out on the range, and a “sheepherder wagon” for overnight guests.

Mr. Hansen says people sometimes think he is full of manure when he describes his job. Most assume the “industry vanished from the planet over a hundred years ago,” he says.

It might have vanished from South Dakota if Mr. Hansen hadn’t learned horsemanship from his grandfather, leather craft from his mother and how to use the welding and woodworking tools in his father’s farm shop. “The resources for a creative craftsman were at my fingertips,” Mr. Hansen says.

In the 1970s, some old-timers shared tidbits about wagon wheel repairs. Soon word spread that Mr. Hansen could restore horse-drawn vehicles, and neighbors started bringing him work. He picked up antique tools from flea markets and custom made the rest. At first, he says, he charged around $10 an hour.

Mr. Hansen’s creations now sell for more than a new Chevy Silverado, with a typical top speed of about 12 miles an hour. He is among a handful of stagecoach builders doing a rollicking business.

Jimmy Wilson in Paradise, Texas, took over his father-in-law’s business and now partners with an Amish “wheel man” in Montgomery, Ind. They specialize in the traditional Concord—“the Cadillac of all the coaches,” Mr. Wilson says, with a sticker price stretching to the low six figures.

Wells Fargo is also a client, Mr. Wilson says. The San Francisco-based firm, the nation’s biggest bank by market share, spends plenty touting its brand’s historic connection to the American West. It has 24 stagecoach replicas, 16 originals and 14 contract drivers who steer the vehicles in parades and corporate events, says Beverly Smith, Wells Fargo’s head of historical services.

The bank wouldn’t disclose how much it spends on its stagecoach appearances, which last year included more than 800 events.

Mr. Tarantino is another high-profile customer. After making a series of films that spanned crime, horror and war, the director turned to Western-style films, beginning with his 2012 film, “Django Unchained.”

For “The Hateful Eight,” Mr. Tarantino and his staff looked at about 20 stagecoach options and settled on a Hansen-built model with a buffalo painted on the side, says Rusty Hendrickson, head wrangler on the film, which is expected to be released by The Weinstein Company later this year.

Some crew members felt the blood-red coach with yellow trimming was “too pretty,” but it held up through miles of rough terrain in Telluride, Colo., where the movie was shot with actors Kurt Russell and Samuel L. Jackson.

Mr. Tarantino didn’t flinch at the $90,000 price tag, Mr. Hendrickson says, and “went whole hog for the stagecoach.”