The disparity of rates among states is stark. In Vermont, the share of children who are fully vaccinated is 52 percent; in Mississippi, it is 6 percent.

“It’s going to be a long slog at this point to get the kids vaccinated,” said Jennifer Kates, a senior vice president at Kaiser who specializes in global health policy. She says it will take unwavering persistence like that of Mr. Kiplinger, whom she knows firsthand because her child attends his school. “It’s hard, hard work to reach parents.”

After the Pfizer-BioNTech vaccine was authorized for younger children in late October, the out-of-the-gate surge in demand lasted a scant few weeks. It peaked just before Thanksgiving, then dropped precipitously and has since stalled. It now hovers at 50,000 to 75,000 new doses a day.

“I was surprised at how quickly the interest in the vaccine for kids petered out,” Dr. Kates said. “Even parents who had been vaccinated themselves were more cautious about getting their kids vaccinated.”