Jeanette Rankin, the first woman to serve in Congress, was a member of the GOP. She was elected to represent the state of Montana in the House of Representatives in 1916 and served two nonconsecutive terms.
The first female candidate to run for the Republican presidential nomination was Margaret Chase Smith in 1964. She did not win the nomination. No woman has ever won the Republican nomination for president. As of July 2015, Carly Fiorina is vying for the 2016 Republican presidential nomination. The only woman who has won the Republican vice-presidential nomination is Sarah Palin, who was John McCain's running mate in 2008.