FOWLER, Calif. — David Cardenas, a mechanic and the mayor of Fowler, knows families in his town want high-quality and free daylong preschool.

But options are thin. A government-subsidized program fills up fast and fits only a small fraction of the town’s 4-year-olds, he said. A private program that closed a decade ago was unaffordable for many of the 6,500 residents of Fowler, a predominantly Latino community of agricultural workers in California’s Central Valley. Otherwise, there are a handful of private day cares.

So Mr. Cardenas recently seized on an unusual preschool alternative that a group from Utah presented to him. “This is something that I have never seen before,” he said. “I wanted to be on the front line right away.”

Mr. Cardenas was referring to a “kindergarten readiness program” for 4-year-olds that takes place almost entirely online. Called Waterford Upstart and run by a nonprofit group, Waterford.org, it has children spend 15 minutes a day, five days a week over the course of nine months, tapping through lessons on a computer. About 16,000 children in 15 states graduated from the program this year, and the Waterford expects to expand the program to a projected 22,000 students by 2020.