As education secretary, Ms. DeVos has limited ability to carry out school choice nationwide, at least without action from Congress. But her previous investments as a philanthropist are paying dividends.

In 2013 and 2014, the most recent years for which financial disclosures are available, several organizations associated with Ms. DeVos invested over $7 million in school choice lobbying efforts in states now considering new bills. Americans for Prosperity, the activist group founded by the Koch brothers, and the conservative American Legislative Exchange Council are also pushing private school choice in statehouses across the country.

The number of American students benefiting from private school choice programs now is relatively small. Estimates by EdChoice, the organization founded by Milton Friedman, the University of Chicago economist who first introduced the idea of vouchers, put the number at 446,000 this year, out of a total school-age population of 56 million. (Three million attend public charter schools, which Ms. DeVos also has championed and which generally do not accept vouchers.)

Advocates say that expanding private school choice would allow parents to remove children from public schools that are not meeting their needs, and note that surveys show parents in existing programs have high satisfaction rates. Competition from private schools, they say, can help public schools improve.

“A lot of families want to have the choice,” Gov. Terry E. Branstad of Iowa said at a rally in January. “We want to make sure all those choices are available, and are as affordable as possible.”

Traditional school voucher programs, which exist in 15 states and the District of Columbia, allow the government to pay private schools, many of them religious, directly. Tax credit scholarships, like the one that helps pay tuition for Ms. Kakayo’s daughter, are a newer and growing form of school choice. They allow individuals and corporations to receive credit on their state income taxes for donations to nonprofits that provide tuition aid to students. Iowa’s program, currently used by 11,000 students, has income limits — $73,800 for a family of four — and the average scholarship award is only $1,583.