Nita Ambani is India’s wealthiest woman, with a $20 billion family fortune and a 27-story home in Mumbai, but she hasn’t been a power player in the art world, until now.

Over the past six months, her Reliance Foundation—named after her husband Mukesh Ambani’s textile and petroleum empire, Reliance Industries—has emerged as a go-to donor for major U.S. museums seeking to mount shows with ties to her homeland. Last fall, her foundation sponsored a show of traditional Hindu paintings at the Art Institute of Chicago. She’s also...