JERSEY CITY, N.J.—At the Scandinavian School of Jersey City, the children don’t have cupcakes at birthday parties, but there is plenty of fruit salad. Instead of meat, eggs and dairy for the 1- to 6-year-olds, there is kale from the preschool’s garden and cashew milk made from scratch.

This sought-after school in an increasingly hip city is unusually committed to the goal of getting children to eat their vegetables. The Scandi School, as it is called, is one of the few vegan preschools in the country.

The school didn’t start out with such a specific diet. But when two children being raised vegan enrolled four years ago, school director Maria Germerud-Sharp didn’t want them to feel left out at meals, so she banned what they couldn’t have.

“One of the core values of the school is building community,” she said. “We all eat together.”

This is a gentle place where 92 children play barefoot to feel a connection to their environment and the air often smells like peppermint or citrus from aromatherapy. Classrooms have bowls of pine cones, seashells and rocks for toys. Some chairs are sawed-off tree stumps.