“The Walton Family Foundation has been deeply committed to a theory of change, which is that we have a moral obligation to provide families with high quality choices,” said Mr. Sternberg. “We believe that in providing choices we are also compelling the other schools in an ecosystem to raise their game.”

The supporters and critics of charter schools, many of them fierce, cannot be easily divided into political camps. Supporters include both Republicans and Democrats, although critics tend to come more from the left. In Washington, where the charter system has strong backing in City Hall, supporters have been more successful than in New York, where opposition from teachers unions and others has kept charter school enrollment to about 6 percent, despite growth in the past decade.

The size of the Walton foundation’s wallet allows it to exert an outsize influence on education policy as well as on which schools flourish and which are forced to fold. With its many tentacles, it has helped fuel some of the fastest growing, and most divisive, trends in public education — including teacher evaluations based on student test scores and publicly funded vouchers for students to attend private schools.

“The influence of philanthropy in terms of the bang for the buck they get is just really kind of shocking,” said Jack Schneider, an assistant professor of education at the College of the Holy Cross in Worcester, Mass.

A separate Walton foundation that supports higher education bankrolls an academic department at the University of Arkansas in which faculty, one of whom was recruited from conservative think tank, conduct research on charter schools, voucher programs and other policies the foundation supports.

Last year, the Walton Family Foundation gave $478,380 to a fund affiliated with the Chicago public schools to help officials conduct community meetings to discuss their plan to close more than 50 schools at a time when charters were expanding in the city.