Is it permissible for my wife to travel with our children (my son 9 years old, and my daughter 6 years old) to India without a mahram in order to enroll them in Islamic schools, given the difficulty of raising children upon the Quran and Sunnah in America?

The basic rule concerning a woman's travel without a mahram (definition) is prohibition and unlawfulness. The mahram must be an adult; a nine-year-old child is not sufficient as a mahram for his mother. However, the prohibition of a woman's travel without a mahram is a prohibition of means, not of ends. It is permissible when there is a need and an overriding benefit, such as preserving children in a better environment for their religion, provided that her mahram accompanies her to the airport at departure and ensures that someone will meet her upon arrival.