The first president women could vote for was Warren G. Harding. Women got the right to vote in 1920, Harding ran for office in 1921. The second president women could vote for was Calvin Coolidge, the third was Herbert Hoover.
Women were guaranteed the right to vote with the passage of the 19th amendment to the U.S. Constitution in 1920.
First Woman to Vote in the United States Under the 19th Amendment (Suffrage Amendment):
Mrs. Marie Ruoff Byrum, August 31, 1920, 7 a.m.
First American Woman to Vote for Her Husband as President:
Florence Harding, Mrs. Warren G. Harding.
(source: Florence Harding by Carl Sferrazza Anthony)
[Added references - McG]