Both her parents were athletic, and young Glenna Collett excelled at such sports as swimming and diving. She learned to play golf when she was 14 and won her first U.S. Women's Amateur championship in 1922. She regained this title five times (1925, 1928–30, and 1935) and was runner-up twice (1931 and 1932). She won the Canadian championship in 1923 and 1924 and the French championship in 1925. In the mid-1920s she won 59 of 60 consecutive matches in tournament play. She was the captain of the American team in the Curtis Cup competition against Britain in 1934, 1936, and 1948.

Collett married Edward H. Vare in 1931 and continued to play golf competitively in women's amateur championships. (Women's professional golf did not become successfully established until after World War II.) She was one of the first six women elected to the Women's Golf Hall of Fame in 1950, and in 1952 the Ladies' Professional Golf Association honoured her by giving her name to the Vare Trophy, awarded annually to the woman professional with the best scoring average.