The Libertarian movement has garnered criticism from progressives and the far-left for their overall refusal to publicly wear the label of radical feminism, but libertarians have a long track record of supporting women's rights. Afterall, the core belief of libertarianism is individual liberty for all people-- and last time I checked, women are people.
Libertarians tend to eschew or denounce left-radical feminism, not for the feminist movement's stance on women's liberation, but rather because of its enmeshment with socialist rhetoric and anti-capitalist organizations. Some libertarians argue that the plight of women has been co-opted or usurped as a means of pushing a socialist agenda by some radical feminist groups, and don't actually advocate for the total liberty of women. (For example, if a woman declares she is a capitalist conservative, the vigor of her support from the new feminist movement seems to wane. Her 'celebrity status' often relies on aligning perfectly with leftist radical feminist orthodoxy.)
Libertarians who strongly endorse women's rights but wish to distance themselves from socialist and communist organizations have coined a new term known as "Libertarian feminism".
Libertarian feminism mirrors much of the philosophies of the original feminist movement; arguing that women should be absolutely free human beings able to pursue their careers, be mothers, be single, own businesses or whatever they choose is the best life for them. Her liberty and right to exist freely is supported unconditionally (no matter if she's a liberal, conservative, Christian, atheist etc.) This is the foundation of libertarianism itself-- to support the freedom and personal choice of the individual.
Politics is a messy playground, but through and through libertarians have always--and will always--defend the inalienable rights of women, and fully support women's rights. The government has no authority to restrict, abridge, compromise or in any way alter the naturally existing rights of an individual, regardless of their sex.
For more information, see What is Libertarianism?
