Is it permissible for a religious, virgin woman, who has memorized the Qur'an and the Sunnah, is educated and beautiful, but is unable to conceive, to search for a husband with the intention of a lawful marriage, given that no suitor has proposed to her?

It is permissible to seek a spouse through lawful means. Indeed, Umar (may Allah be pleased with him) offered his daughter Hafsa to Uthman (may Allah be pleased with him). Likewise, a woman came to the Prophet (peace be upon him) and offered herself to him. These hadiths indicate the permissibility for a woman's guardian, or the woman herself, or someone she deputizes, to offer her to a righteous man for marriage.