In some of the versions of Rāmāyaṇa (not Vālmīki Rāmāyaṇa), the famous vānara Hanumāna who was a brāhmacāri, had a son miraculously. One of the tales goes like, after Hanumāna burnt the entire Laṅkā with his tail, due to tremendous heat emerging out of his body, he dived into the sea to cool himself. It's said that, in the process, a drop of his sweat fell inside the mouth of a large fish (probably crocodile as the name suggests) and that impregnated it. In some other versions, it was Hanumāna's reproductive fluid that fell along with the sweat inside makara's mouth. That's how Hanumāna's son named Makardhvaja was conceived.
In the bhakti tradition, Nābhādas, the author of Bhaktamāla, is described by Priyādāsa in his commentary Bhaktirasabodihī as Hanumāna-vaṁśī, but he never defines the term, and it has had many interpretations. Rūpkalā, in his commentary named Bhaktisudhāsvādu, lists many interpretations of the term explains that Nābhādas was born from a drop of Hanumāna's sweat. He narrates the story of how Nābhā-jī came to be born in an unusual manner. One time, Śiva was instructing Hanumāna in Yoga. Due to Hanumāna's great mental effort, a drop of sweat fell from his body. This drop was caught by Śiva in a container, and in order to increase bhakti, he threw it down to earth where it became Nābhā-jī. That's one interpretation of Rūpkalā, as to why, Nābhādās is called Hanumāna-vaṁśī, and showing him to be an ayonija i.e. not born out of a woman's womb. (Hare, 2011)
The Kṛttivāsī Rāmāyaṇa, is one of the later versions of Rāmāyaṇa that was composed by the fifteenth century Bengali poet Kr̥ttivāsa. In that, there is another episode, when the famous king Dilīpa of the Ikṣvāku dynasty died, there was no heir to the kingdom. Śiva appeared and commanded the two widowed queens to make love together and blessed them that they would have a child that way. The two queens executed Śiva's order and one of them conceived a child named Bhagīratha, the famous ancestor of Rāma.
So, Bhagīratha was born by the sexual congress of the widowed wives of the late king, and this episode is not there in the Vālmīki Rāmāyaṇa. (Bose & Bose, 2013)
In the Kannada retelling of Rāmāyaṇa, Rāvaṇa (Ravul̥a), with his queen Mandodari was childless. So, Ravul̥a goes to the forest to perform austerities and meets Śiva, disguised as a yogin. Śiva gives him a magic mango and asks him to share it with his wife. Ravul̥a agreed, but Śiva said that he wouldn't do as said, and then will bear the repercussions of his own disobeying of the command. On the way to Laṅkā, Ravul̥a felt extremely hungry that he ate the mango himself, thereby fulfilling the prediction of the yogin. Thus, he became pregnant, and each day of pregnancy was equivalent to a month. At the ninth day, he sneezed and out came a little baby girl. The oracles said that the girl would be Ravul̥a's death and so he put her in a box and buried her in a field faraway.
The traditional bards who narrate this tale say that Sita was called
Sita because she was born when her father sneezed (in Kannada, sita
means sneeze). And in Vālmīki Rāmāyaṇa, her name is Sītā (as in furrow) because she was found in a furrow when her foster father Janaka was ritually plowing the sacred fields. Thus, in the Kannada folk retelling of Rāmāyaṇa, Sītā is ayonij. (Ramanujan,1991)
Mahābhārata has an instance of an Ikṣvāku king named Yuvanāśva (who was already married) who gave birth to his son Māndhātā by himself. ( Sections CXXVI, LXII, XXIX )
The story went like this, Yuvanāśva who had retreated to the forest, observed a fast once. But unfortunately, he suffered from terrible hunger as result and from a hermitage drank/ate some sacrificial water/butter from the jar/pot. Apprently, the water/butter was not ordinary but a magical type which could make the one who drank it pregnant, and bestow them with a godly son. That's how Yuvanāśva became pregnant and gave birth to a godly son.
Droṇa, the teacher of Pāṇḍavas and Kauravas, was pot-born and untouched by a mother's womb. Sage Bhārdavāja had gone to river Gangā for offering havis when he saw an apsarā Ghr̥tācī draped in wet clothes with water dripping from them. When a gush of wind drew away her clothes, the sage was aroused and lost to ejection of some reproductive fluid, which he then put in a water pot (droṇa). Thus, Droṇa was born from a water pot and had no biological mother, as you mentioned. Droṇa's brother-in-law Kr̥pa and Droṇa's wife Kr̥pī, too were not born of mother's womb. Their father Śaradvat, son of sage Gautama, was aroused upon seeing an apsarā sent by Indra to upset his concentration and practice. And as a result, he lost self-control and ejaculated. The semen fell on a reed and split in two halves, from which born twins, Kr̥pa and Kr̥pi.
Even in the accounts of these two motherless births of Droṇa and Kr̥pa-Kr̥pi, women did play an indirect role of initial motivators, that of facilitating arousal. There are some birth accounts in Mahābhārata, where women don't play a role at all. These births have been described as ayonija or non-uterine.
As is known, Drupada out of desire to avenge Droṇa for hs humiliation, scouted and found a suitable priest to perform a yajña for the generation of a son, who could slay Droṇa, At the end of havana (oblation), the priest told the queen 'a twin for you has appreared' and invited her to eat the havis (offering). The queen asked the priest to wait for a while so that she could go and wash her mouth and have a bath. But the proud priest refused to wait and put the havis into the ritual fire. From the fire of the sacrificial altar, sprang a boy, Dhr̥ṣṭadyumna and the princess Kr̥ṣṇā, better known as Draupadī, both having terrible heavenly announcements associated with their fate and purpose.
In Bhāgavata Purāṇa (IV.14-15), we see another case of ayonij, i.e. of Pr̥thu. Pr̥thu was churned out of the body (arms) of the late king Veṇa by the sages, so he was born without any female involvement.
In popular oral tradition, Rādhā, the beloved of Kr̥ṣṇa, is believed by some devotees to be ayonija, where they refer a folktale of her as a child being found laying on a lotus leaf by the wife of Vr̥ṣbhānu (in some versions by Vr̥ṣbhānu himself), who became her foster parents later. The Garga Saṁhitā is sometimes wrongly cited to support this, since I.8.6 clearly mentions Rādhā's entry into the womb of Vr̥ṣbhānu's wife. In Kr̥ṣṇa-Janma-Khaṇḍa (XVII. 146-147) of Brahmavaivarta Purāṇa though, Rādhikā (Rādhā) is born as the daughter of Kalāvatī and Vr̥ṣbhānu, but not out of the female womb as clearly mentioned in the verses. Thus she is ayonij, as per this text.
Popular tradition also talks about the birth of Ayyappa, where in some versions he is said to have been born by the union of Mohinī (the feminine form of Viṣṇu) and Śiva. This example can be problematic because Viṣṇu is considered a male but also has a feminine form in this case, so would he be considered a woman or man? There is another problem which is present in the conception of children by deities. It's hard to restrict the high ranking Purāṇic deities (trīmurti) to specific genders, for e.g. Śiva is seen in certain traditions as being both feminine and masculine, such as in his form Ardhanārīśvara. For e.g., Brahmā who with his mental power alone, without any involvement of women, produced his mind-born sons (mānasaputra) in the Viṣṇu and Bhāgavat Purāṇa. The problem with this is that we can't clearly call Brahmā or Viṣṇu or Śiva, no matter their form, as exactly male or female because they are beyond the gender categories of ordinary humans, and thus they are often personified in traditions with both male and female forms or genderless.
Most of the devas or some of those blessed with divine powers are beyond gender forms and so, many of them have the ability to take the forms of man or woman at will. Let there be a case, where a supposedly -male deity takes the form of a woman and has intercourse with a man, then they can have a child (for eg. Sugrīva was born this way). But would it still be considered a correct example? Because the deities don't have the limitation of ordinary hetro-sexual humans in which child can only (usually) be produced with the aid of the opposite sex.
Some of the examples from the epics given, are of the children of married people, or of women playing an indirect role (of motivators) in the birth (such as Droṇa). Makaradhvaja and Nābhādāsa are the only example given where they were born from womb. Although there is belief in bhakti traditions that Nābhādāsa is ayonija, but that is just a belief, since he is an actual historical person. As for the rest of the examples, none of these are born of actual intercourse or a female's womb, the only issue is that even though in some examples no woman was involved in the conceiving of their child (such as in the case of Yuvanāśva, Ravul̥a, Draupadī etc.), most of the men were married. The example from Kṛttivāsī Rāmāyaṇa of women themselves making love and having a baby without a man is interesting and unique even though it's not a correct example for the question, so it deserved mention according to me, on it being quite a peculiar birth. These are the only (semi) examples I could give. I hope it's not useless.
References
Hare, J. P. (2011). Garland of Devotees: Nābhādās’ “Bhaktamāl” and Modern Hinduism. [Doctoral dissertation, Columbia University]. ProQuest Dissertations Publishing. https://doi.org/10.7916/D8VH5TN9
Nābhādās, Priyādāsa, Rūpkalā (1962) Bhaktamāla of Nābhāji with the Ṭīkā of Priyā-dāsa and with the commentary entitled with Bhaktisudhāsvādu of Śrī Sītārāma-Śaraṇa Bhagavān Prasād Rūpkalā. (3rd ed.) Naval Kishore Press. Lucknow
Bose, M., & Bose, S. (2013). Appendix 2: Kṛttivāsī Rāmāyaṇa. In A Woman's Ramayana: Candrāvatī's Bengali Epic (pp. 103-111). Routledge. https://doi.org/10.4324/9780203441480
Ramanujan, A.K. (1991). 2. Three Hundred Rāmāyaṇas: Five Examples and Three Thoughts on Translation. In P. Richman (Ed.), Many Rāmāyaṇas: The Diversity of a Narrative Tradition in South Asia (pp. 22-49). University of California Press. https://doi.org/10.1525/9780520911758-004
Ganguli, K.M. (1883-1896). The Mahabharata of Krishna-Dwaipayana Vyasa . Munshiram Manoharlal. https://www.wisdomlib.org/hinduism/book/the-mahabharata-mohan
Chakrabarti, A., & Bandyopadhyay, S. (Eds.). (2014). Significance of the Early Parvans: Modes of Narration, Birth Stories and Seeds of Conflict. in Mahabharata Now: Narration, Aesthetics, Ethics (pp. 37-56). Routledge India. https://doi.org/10.4324/9781315662114
Tagare, G.V. (1900). The Bhagavata Purana. (J.L. Shastri, Ed.) Ancient Indian Tradition and Mythology Series (Vol. 7- 11). Motilal Banarsidass. https://www.wisdomlib.org/hinduism/book/the-bhagavata-purana
Goswami, D. & Dasa, K. (2006). Śrī Garga Saṁhitā. Rupanuga Vedic College. https://www.wisdomlib.org/hinduism/book/garga-samhita-english
Nagar, S.L. (2005). Brahmavaivarta Purāṇa. (R. Chaturvedi, Ed.) Parimal Publications. https://archive.org/details/brahma-vaivarta-purana-all-four-kandas-english-translation
