In theory, the first First Lady who would be able to vote was Edith Wilson, as the 19th Amendment to the U.S. Constitution was ratified on the 18th of August 1920. However, "Edith Wilson remained steadfastly against the Suffrage Movement. When President Wilson arrested the suffragettes who were at the entrance to the White House, Edith insisted that they remain in jail until they repented and promised not to protest again." [ Source ] Whether or not this meant she didn't vote, we cannot say for certain, but it does seem unlikely.

It's often cited that the first First lady to vote was Florence Harding, who voted for her husband in 1920, however she was, obviously, not the First Lady by then. [ Source ] It is, however, very cool that there is a photo of her voting. As her husband died whilst in Office, she ceased to be First Lady before she could vote in the next election.

Therefore, in actual fact, the first First Lady to vote whilst being First Lady was Grace Coolidge, who, "On 30 October 1924, she did participate in a small but significant politically non-partisan gesture to encourage women to exercise their right to vote for the second time since passage of the 18th Amendment. Having her personal desk brought out onto the South Lawn, she sat and filled out an absentee ballot while press photographers recorded the simple act and disseminated it through newspapers." [ Source ] Again, there is a photo of her doing this.