May Sutton was born in Plymouth, England, but when she was six years old, Sutton's family moved to a ranch near Pasadena, California. It was there that she and her sisters played tennis on a court built by her father. As young ladies, May and her sisters, Violet Sutton, Florence Sutton, and Ethel Sutton, dominated the California tennis circuit. In addition to being accomplished tennis players, the girls were excellent basketball players. May, Florence and Violet were all on the Pasadena High School basketball team, which went undefeated in 1900. In 1904 at age 18, May Sutton won the singles title at the U.S. Championships. She also teamed with Miriam Hall to win the women's doubles title and came close to making it a clean sweep by advancing to the mixed doubles final.
In 1905, she became the first American and first non-British woman to win the Wimbledon singles title when she beat British star and reigning two-time Wimbledon champion Dorothea Douglass Chambers. She did it while shocking the British audience by rolling up her sleeves to bare her elbows and wearing a skirt that showed her ankles. For the next two years, she and Chambers met in the final, with Chambers recapturing the title in 1906 and Sutton winning it back in 1907.
May Sutton was the 1908 Rose Parade Queen in Pasadena.
In 1912, she married Tom Bundy, who was a three-time winner of the men's doubles title at the U.S. Championships, and semi-retired to raise a family. However, in 1921 at the age of 35, she made a comeback and became the fourth-ranked player in the U.S. In 1925, she was a women's doubles finalist at the U.S. Championships and, although almost forty years of age, her game was strong enough to be selected for America's Wightman Cup team. She was a Wimbledon quarterfinalist in 1929 at the age of 42, which was the first time she had played Wimbledon since 1907. In 1928 and 1929, she and her daughter Dorothy Cheney became the only mother/daughter combination to be seeded at the U.S. Championships. Her nephew, John Doeg, won the U.S. Championships in 1930, and in 1938 daughter Dorothy won the Australian Championships.
Eight-time U.S. National Championship winner Molla Bjurstedt Mallory indicated that Sutton was the best player she had met. "Her drive was the fastest and the ... most difficult ... to handle, because it dove suddenly to the ground and then jumped up unexpectedly with queer curves. When she could keep her drives near the baseline, they either forced me back farther than I had been accustomed to play or compelled me to make errors. She was also strong overhead when she came to the net and altogether had more power and effectiveness than any other woman tennis player of her time". Sutton played with an extreme Western grip and had a powerful topspin forehand that made the ball dip and bound high.