In 2015, Nevada passed an empowerment scholarship bill that would have allowed any former public school student access to use an account, but last year, the State Supreme Court halted the program, saying its financing mechanism would harm public schools. The program is on much firmer legal ground in Arizona, where the state’s top court declared the accounts constitutional in 2014.

This year, about 3,500 Arizona students, the majority of whom have special needs, are participating in the program. The average size of an account is $5,700 per year for children without disabilities and $19,000 for children with them. The funds are distributed via debit cards.

According to a 2016 report on the program from EdChoice, an advocacy group that supports private school choice, 83 percent of E.S.A. funds are spent on private school tuition, 7 percent are spent on tutoring and less than 1 percent are spent on online education. A third of recipients are using their accounts to pay for multiple educational services.

In legislative negotiations, skeptics of the expansion were able to secure several compromises. Although all public school students will be eligible to apply for an account, enrollment in the program will be limited to about 5,500 new students each year, eligible on a first-come-first-served basis. Total enrollment will be capped at 30,000 students in 2022. All private schools that accept E.S.A.s will be required to test account recipients annually, using a nationally recognized standardized test. But only schools serving over 50 scholarship students will have to report their students’ scores publicly. Unlike public schools, they will not have to report on the performance of subgroups such as racial minorities, low-income students or those learning English.

Advocates for education savings accounts say they allow children from low-income families to escape failing public schools. But an investigation by The Arizona Republic found that 75 percent of current E.S.A. recipients in the state previously attended relatively affluent, high-performing public schools.

The program has been especially popular with the parents of disabled students. “The public school options are not working for those families,” regardless of income, said John Schilling, chief operating officer of the American Federation for Children, a national advocacy group that lobbied for the Arizona expansion. Ms. DeVos is the group’s former board chairwoman.

An Arizona state senator who opposed the bill, Juan Mendez, a Democrat, said the amount of money in the accounts was too little for many of his low-income Latino constituents to afford a high-quality private education. He worries, he said, that affluent white parents will use the accounts to flee public school systems that are becomingly increasingly Latino each year.