Have any orally-transmitted myths survived throughout the centuries in such a form that we could identify any of the ancient philosophers of Classical Greece and Rome in them? I'm interested in finding out how much could their philosophies actually affect thinking of people in the Middle Ages.
I think you want to start by reading a little Dante.
Here I saw Socrates and Plato, who in front of the others stands nearest to him; Democritus, who ascribes the world to chance; Diogenes, Anaxagoras, and Thales; Empedocles, Heraclitus and Zeno; and I saw the good collector of the qualities, Dioscorides, I mean, and I saw Orpheus, Tully and Linus and moral Seneca; Euclid the geometer, and Ptolemy, Hippocrates, Avicenna, and Galen, and Averrhoes, who made the great comment. - Inferno, Canto IV.
Dante clearly expected his audience to know who these people were. Which tells you that as of about 1300AD, people had some smattering of knowledge - and not just the educated monks, but the general audience that Dante was writing in vulgar Italian for. Dante assumed the burghers knew not just of the names of the philosophers, but a bit about their most influential ideas.
We can only guess how much of this written literature was influenced by the oral literature of the era, or how it seeped into the oral culture that followed. But Dante wrote in verse for a reason - it was written to be read aloud and perhaps to be memorized and passed on orally. It seems hard to imagine that Dante's works were isolated from the oral literature of the time.
Edit: Now that the OP question has been reworded for clarity I will tip my hat to pokep as being closer to answering the intent of the inquiry. It was not clear to me originally where orality and myth came into the picture with literature in the question. Nonetheless, I'll leave my answer here in case it helps anyone.
I'm afraid you're probably not going to get away without doing some research of your own on this, but perhaps some pointers could help. Firstly, since you give such generous leeway with respect to the period of time under consideration (from the lifetimes of the ancient philosophers up to the Middle Ages) I feel at liberty to jump around a little bit.
Firstly, from your title alone ("How much of what we know about ancient philosophers comes from oral literature?") I immediately thought of the matter of Socrates and what we know of him and how we know it. We have no writings from Socrates himself, and are dependent largely on Plato for telling us what Socrates taught. That in itself presents us with a dependence on orally transmitted details about a person of a truly ancient provenance (almost from the get go). So oral transmission of knowledge about a person need not be something long after the fact and also need not be inherently suspect or inaccurate.
It seems ironic, however, that the historicity (existence) of Socrates as a person and the essential content of his teachings are rarely called into question, despite us having no written records from him personally; yet all the while some "mythicists" call into the question the historicity of Jesus of Nazareth and his essential teachings (which also relies on sources that provide orally transmitted details from early witnesses who knew him personally, rather than from any direct writings by him). The same can be said for Siddhārtha Gautama (Buddha). There are good reasons to regard them all as real historical figures whose teachings have reached us accurately by their ancient witness-bearers who wrote them down.
To take one ancient example from the Hellenistic world, the fields of New Testament studies (in the case of Jesus and early Christian history - as Koine Greek speakers and/or writers), Orality, Textual Transmission, and Historiography are all even disciplinarily related - resting upon the same historiographical foundations and inquiries - to this same question you raise about orally transmitted details of ancient figures and their teachings. The questions and methodological principles for analyzing such orally transmitted accounts (especially in such a largely oral culture) applies equally to New Testament studies and the study of any Greek or Roman literature or historical sources in general. So studying orally transmitted details about Greek Philosophers will involve the same interdisciplinary considerations to determine historicity.
The field of study on those topics of orality and historiographical textual studies is vast, and I have seen much discussion of it in NT studies (which is why I mention it), and no doubt broader studies on Greek literature and historical sources may be found which would apply to Greek Philosophers. I would just recommend spending time on a search engine, journal aggregate sites like JSTOR (which now provides some limited free access), or book catalogues such as WorldCat searching for keywords like "Orality" and "Greek literature" or "Greek History" (Substitute "Hellenistic" for "Greek" where appropriate) and start where you can.
Further, aside from such foundational matters, if you wish to find a good compendium of ancient knowledge about all manner of elements of Greek and Roman culture and language as understood by a later civilization, the Byzantines, then I recommend doing some research on the Suda. This may help bridge the gap from classical times to the Middle Ages, as you discussed. As Wikipedia states the Suda "is a large 10th-century Byzantine encyclopedia of the ancient Mediterranean world". It is part lexicon and part encyclopedia but presents a broad understanding of knowledge of the classical world, and may provide you some clues for what you endeavor to research. I often see it cited in academic literature when someone is debating the meaning of Greek words in classical literature (often as applied to exegetical analysis of Greek texts), but it also provides some details of ancient knowledge and facts as understood at the time. As the Wikipedia article states:
The articles on literary history are especially valuable. These entries supply details and quotations from authors whose works are otherwise lost.4
One Byzantine churchman and scholar who wrote widely on Greek classics and also made reference to the Suda was Eustathius of Thessalonica. In the Encyclopedia of Ancient Greece by Nigel Wilson (pg. 60) it says:
Noted Classical scholars such as Michael Psellos and John Italos in the 11th century and John Tzetzes and Eustathios of Thessalonica in the 12th did not confine themselves to the copying and editing of ancient manuscripts but became the self-appointed exegetes of the Classical tradition, writting essays and commentaries in a broadly "Classical" style, albeit from a Byzantine perspective, and using ancient texts to reflect upon contemporary and often ecclesiastical issues. During the Palaiologan period (13th century on) there was a revival in Classical philology and many Greek texts owe their survival to the efforts of Planudes in the 13th century and Triklinios in the 14th.
Without you narrowing down your inquiry further though it is tough to know what specific areas to hit upon. However, those topics about Socrates, the Suda, and later Byzantine Greek scholarship were the first things to come to mind when I read the OP. Perhaps this will be of some assistance.