It is not permissible for a woman to travel without a mahram, even if she is travelling for the obligatory Hajj or obligatory ‘Umrah, because of the report narrated by al-Bukhaari (1729) and Muslim (2391) from Ibn ‘Abbaas (may Allaah be pleased with him) who said: The Prophet (peace and blessings of Allaah be upon him) said: “No woman should travel except with a mahram, and no man should enter upon her unless there is a mahram with her.” A man said: “O Messenger of Allaah, I want to go out on such and such a campaign, and my wife wants to go for Hajj.” He said: “Do Hajj with her.”
With regard to the naafil Hajj, a woman is not allowed to travel for that without a mahram according to the majority of scholars, and some narrated that the scholars were unanimously agreed on this point.
The fact that a group needs a female doctor to travel with them and that there is no time to look for another doctor, does not make it permissible for you to travel with them without a mahram. How can a person draw closer to his Lord by doing Hajj or helping others by means of disobedience?
We ask Allaah to guide you and make you steadfast.
And Allaah knows best.