With her laptop open and credit card out, Allyson Mendoza watched the clock flip from 7:59 to 8:00 a.m. “Register now” popped up on her screen.

The mother of three had set timers and reminders for this moment weeks in advance of the March deadline. By 8:02 a.m., she had scored spots for her two oldest children at Hummel Day Camp, the wildly popular city-run day camp held for the last 70 summers in Omaha’s Hummel Park. She had done so with mere minutes to spare.

“It’s the most stressful 10 minutes of my life, and I’m a lawyer,” Mendoza said.

The Mendoza children and 2,000 othe...