Generally no: private K-12 schools do not receive federal education money directly, although their students and teachers can benefit from federally funded services.
Under the Elementary and Secondary Education Act, as amended by the Every Student Succeeds Act of 2015, local public school districts must provide “equitable services” to eligible private school students, teachers and, in some cases, families. These programs include Title I, Part A help for low-achieving students who live in high-poverty public school attendance areas, Title II teacher training, Title III English learner support, and Title IV enrichment programs such as 21st Century Community Learning Centers. The money set aside must be equal on a per-pupil basis to what public school participants receive. However, a public agency controls the funds and keeps title to any materials and equipment bought with them, and the services must be secular, neutral and supplemental. A student who attends private school through a state voucher or other school choice program is still eligible for these services.
Image: Lyndon Baines Johnson Department of Education Building, Washington, D.C., by APK, CC BY 4.0, via Wikimedia Commons.