Jul 28, 2015 2:03 PM

It was a normal Saturday in a New Jersey ice cream shop two springs ago—families shuffling in and out, ice cream cascading over the brims of cones and onto children’s plump cheeks—when all attention focused on a faint rumbling a few hundred yards down the road.Speculation grew as the increasingly ferocious roar filled the ice cream parlor. And then a bright yellow 1981 Kenworth cab-over flat-front truck barreled around the corner.

The driver of truck was not a bearded, burly middle-aged man, wearing a sleeveless flannel button-up and Timberland boots, as one might stereotype. Rather, the man behind the steering wheel was a Brooklyn-based DJ and former BMX pro, Jeff Mayer, with his petite wife, Angela De Vincenzo, and 4-year-old son, Luca, in the passenger seat. Hungry for a cold, mid-afternoon snack, they descended a ladder and entered the shop.

Two years ago, the couple had taken a huge leap—probably against their better judgment, they acknowledge—and bought Luca a seven-ton 18-wheeler. Born with a cleft palate, the boy had developed speech at a later age than most children, but it never squashed his love for trucks, like many of his peers. In fact, it was around trucks that Luca became expressive and outgoing, his parents said.

“Luca was and still is obsessed with trucks, and one day he said, ‘Papa, I really want a truck,’” Mr. Mayer said during an interview, seated with his family in the trailer, parked at the Hayground School in Bridgehampton. “And I was, like, ‘Who in the world would get their 4-year-old an 18-wheeler? I started to look on Craigslist and Facebook pages and, one day, I found this one, and the price and everything were perfect. Actually, it was the exact amount of my bonus that year.”

Mr. Mayer and Ms. De Vincenzo interpreted the coincidence as fate and bought the truck, which they now store in New Jersey at Mr. Mayer’s parents’ home. But the attached trailer, and the idea that kick-started Luca718—the family’s new business, named after their son and area code—are fairly recent acquisitions.

So new, in fact, that Hayground is their first stop.

“We bought the trailer this past February or March, and it had about $5,000 of work that had to be done before we could come out here,” Mr. Mayer explained, where they will live inside the truck for the summer and teach campers about art, blocks and trucks from the back of the trailer.

“Besides teaching the children about blocks, I’ve actually been able to help several of the children with reading,” said Ms. De Vincenzo, who works as a learning specialist in Brooklyn. “Between all the activities the children can do here, I can get them focused on something and then help them with issues they struggle with. Just yesterday, a mother called me and told me that her child recited all of the vowels to her during lunch—something that she has been trying to learn for months.”

Mr. Mayer now focuses on interior and clothing design, in particularly their Luca718 merchandise, as well as his original passion—spinning tracks as a DJ. He revamped the interior of the renovated 48-foot trailer, which now includes a lounge area stocked with a record player and countless records, a large truck-shaped bed, and an art gallery for the children. With gigantic doors that swing open on both sides of the trailer, the cross breeze is unmatched. It is easy, while lying on the leather sectional, to forget that it is a truck trailer and not a modest beach cottage.

The side of the truck reads, in massive letters, “Keep on Truckin’!” “It’s kind of our motto,” Ms. De Vincenzo said.

“We hit a few roadblocks before we were able to get the trailer and truck out here and running,” Mr. Mayer laughed, recalling a team of mechanics welding the underbelly of the truck in the wee morning hours before leaving for Hayground. “And each time we hit a bump in the road, the saying just kept coming back to us.”

Hayground is the first step on what the family hopes will be a very long journey. “We are calling it our East End tour from Brooklyn to Montauk, but for right now the only thing we really have scheduled is here at Hayground,” Ms. De Vincenzo explained. “Ideally, we want to be able to take ‘Big Mama’ and the trailer around to different schools and camps, and reach a wide audience of children across the country. We are doing something really special, and this rig is literally the vehicle and a workshop by which we can create and empower children to think together.”

“Luca is the reason we are doing all of this,” Mr. Mayer said. “His teachers told us this year, he is so confident, and it’s because of this truck. And that’s what we want to do for as many kids as we can: give them confidence and empower them to think, create and make their own decisions and explore outlets they otherwise wouldn’t be able to.”