Whenever a question is posed, “Does Jewish law permit?” we enter the realm of it depends what we mean by Jewish law and who we ask.
As Rabbi Marc Angel has said, “There is not one pesak halakhah (Jewish legal decision), rather a range of piskei halakhah (plural).”
I am in the fortunate position of not being a posek halakhah (Jewish legal decisor). So, I will not give a final answer to this question. As they say, it depends who you ask.
The idea of an unmarried woman giving birth through whatever means is in traditional terms, unprecedented and obviously unacceptable.
But from the standpoint of modernity and the emphasis upon the individual and personal fulfillment, it is understood why such a question would be asked and why an individual might desire to give birth and nurture a child.
Having the technology to bring about a result of a human birth which is considered a “berakhah”— a blessing seems to make sense and since others are doing it, “so why shouldn’t I?”
But, as with anything else, the personal fulfillment aspiration does not exist in a vacuum and the question relates to more than what the self wants, rather “Does Jewish law permit in vitro fertilization?”
For a woman desiring an answer to this question, she must turn to an actual rabbi who is competent in the field of Jewish medical ethics and the latest decision-making in halakhah. She can then ask a she-ailah—a question requiring a teshuvah—response.
We must remember an overarching principle that was taught to me by the great Rabbi Dr. Moshe Tendler, widely respected especially in this field. He said something to the effect that Judaism is on the side of natalism. In other words, the Torah favors doing whatever is necessary and available in order to bring children into this world.
Rabbi Tendler , the scholar that he is and the scientist that he is, would not, I believe, go beyond the limits of the traditional definition of Jewish marriage—a man and a woman.
As I have stated at the outset, I will not make a decision on my own about in vitro fertilization for an unmarried woman, since I am not in a rabbinic position to make such determinations.