Yes, there's a story, but not a very cohesive one.
Wikipedia cites Virgil in the Renaissance's chapter, Virgil with an i, which is a thorough examination of the name's orthographic history:
Vergil or Virgil? The correct answer, as any Latinist can tell, is that the poet spelled his own name with an e: Vergilius. Where did the i come in? A scribe, taking dictation, hears the word Vergilius (which he has never seen before), and writes down the word Virgilius, which sounds right and starts in a familiar way, with the Latin word for man. Or perhaps he thinks of virgo, "virgin," because Virgil was nicknamed Parthenias, “the virgin"; or of virga laurea, because according to legend Virgil's mother dreamt that she gave birth to a laurel branch; or of virga populea, because Virgil's mother is said to have planted a poplar branch to mark his birth-place.' There were other stories as well. What concerns us here, though, is not how the confusion began, but why it has persisted.
In fact, the question was settled long ago, by a man who is universally recognized as the greatest classical scholar of the fifteenth century. His parents named him Angelo Ambrogini (1454-94), but his readers knew him by his Latin surname, Politianus, on account of his birthplace, Montepulciano.
Where possible, Poliziano also used non-literary texts to corroborate the readings of his literary manuscripts, and this was how he arrived at the correct spelling of Virgil's name. "On the question," he begins his little essay on the subject, "of whether we ought to say 'Vergilius' or 'Virgilius,' I observe that the learned are still walking around in circles (ambigi)." This is a small point, he admits, but he begs his readers not to get impatient if, like a man wandering on the beach, he sometimes picks up a random shell for his own amusement. "Virgilius," he knows, is the spelling that most people use, but "certain records, of great antiquity" have convinced him that "Vergilius" is preferable. The first of these is not a document but an inscription: "In Bolsena," he says,
you will find inside the chapel of Christina the Virgin a certain marble table which is in front of St. Peter [of Prague]'s altar; here, in old letters that have almost rubbed off, the word VERGILI is legible. In Sutri you will also find the name "Vergilius" written in the same way on a stone table ... Both of these we have inspected closely, and not without several people present. For in the examination of these ancient things we did not wish to be ear-witnesses only, but eye-witnesses as well. The name is also written with an e in the original of Justinian's Digest that is preserved in the official archives of Florence. It is the same in the Vergil that is displayed in the inner chambers of the Vatican library, a volume of astonishing antiquity and written out in large characters. [...] [The same spelling] is also found in a codex of St. Augustine's City of God in the Medici public library and in a codex of Columella from the same family's private library, both of which codices are written out in litera langobarda. Then there is a very old book of Seneca's letters, access to which was provided to me by Nicolaus Micheloctius, a man of discriminating intellect, from the private things of Lorenzo de' Medici, along with many other volumes of venerable antiquity. On several occasions we have shown either what these things contain, or what they leave out, sometimes to Jacobus Pratensis, our sober companion and research assistant, and sometimes to others who would listen. [In Fra Giovanni Giocondo's collection], I also come upon the report of two epitaphs which are found among the marbles of Rome, just as Giocondo indicates: TI. VERGILIUS DONATVS and again C. PAPIRIVS CESTUS VERGILIAE OPTATAE VXORI SVAE BENE MERENTI DE SE. And yet, although it is confirmed by records of age-long antiquity (a sufficient defense, I should think), yet reason and the thing itself also bear witness. The term Vergiliae ["Pleiades"] is said to come from verae stelle ["true stars"]; the poet's name is correctly derived in the same way, from these very stars, or even from ver [spring] - not from virga laurea, which is just an old wives' tale found in a not very respectable author, and therefore false. Indeed, many people were named "Vergilius" before the poet was even published. To sum up, where there is either no weightier evidence or no stronger reason, the question is obviously to be settled on the basis of ancient usage (uetusta... consuetudine) rather than recent ignorance (noua inscitia), which is where we get this hodgepodge element coming into our languages. Indeed, we most of us in recent times suffer from a clouded vision of the truth (caligamus ad ueri conspectum). But to follow up on what's left, the name Verginius is also closely allied to this one, and when we recently looked at the carved marble basin in front of the chapel of Santa Maria Maior in Rome, we did not find Virginius.
Dang. The section goes on to confirm Polizano's findings, then continues by reporting Europe's reaction:
Europe yawned - and went on spelling the name as it had always done, with an i. Indeed, it is unusual to find it spelled otherwise until the latter half of the nineteenth century; and it is still spelled with an i in most of the European vernaculars. What happened? We dwell on this trifle - what Poliziano calls his seashell - because it illustrates some larger points about the peregrinations of information, the publication of classical texts, and the general character of Virgil scholarship in the fifteenth and sixteenth centuries.
Poliziano was, by all accounts, a prodigy and he was not the sort to hide his light under a bushel. Yet in this one thing, Europe closed its ears to him. In trying to understand why, we should not underestimate the force of inertia, combined with sloth and compounded with reverence. Literate men, teachers and students alike, had been spelling Virgil with an i since the fourth century AD; by the time Poliziano tried to break it, the habit was more than a thousand years old.
But there were more reasons than just habit. One of them was Poliziano's novel method of getting the word out. In previous generations, the normal format for academic publishing was the commentary, which proceeded through a text line by line and often word by word. The most basic commentary was linguistic and simply provided vocabulary; syntax was also explained where necessary, along with references to history, geography, and whatever else the commentator happened to know.
And then, crushingly, the preview cuts off. We have no idea how Polizano "got the word out", and why it failed at creating his intended splash. So, we move on.
The matter appears to have motivated someone to write a thousand+ word letter to The Nation in 1907, of which selected snippets are below (don't worry, the author of the letter, a guy named Francis Kelsey, does cite a bunch of sources):
This spelling [Vergilius] is attested both by the concurrent testimony of the older manuscripts of the poet and by the consistent use of the name in inscriptions as Vergilius from the second century before Christ until the fourth century of our era. By the sixth century Virgilius had come into vogue, for reasons which it would not be pertinent to our present inquiry to analyze; and when the vernacular languages of modern Europe began to be reduced to writing the late Latin form Virgilius was commonly used; hence, forms of the name in Italian, Spanish, French, German, and English came to have i as the vowel of the first syllable.
In the following century [the 16th], the two spellings of the Latin name appear side by side; there was a rivalry between Virgilius transmitted from the Middle Ages and Vergilius restored by scholars. Not far from one-third of the sixteenth-century editions of Virgil in the collection of the British Museum have the latter spelling in their titles, and in the same period Vergilio was used in several Italian translations. After the sixteenth century, the restored spelling lost ground rapidly, no doubt on account of the influence of the vernacular forms; it became increasingly rare until by the end of the eighteenth century, it had almost entirely disappeared both in editions of the poet and in the writings of scholars.
Altogether different is the question of the spelling of the name in English. It would be a waste of time to bring forward proof to show that from the beginnings of the English literature until now Virgil has been the accepted literary form.
The convenience of classical students would, of course, be subserved by the general adoption of the spelling Vergil; how awkward it is to be obliged to write the Latin form with one spelling and the English with another! Many of us hoped that the revised spelling would be acceptable and would generally prevail. After four decades, however, we must, I think, frankly admit that the effort to make current in English the corrected spelling Vergil is a failure; the tendencies of our vernacular are too strong to be shaped by the protests and example of a group of specialists. But of classical scholars by no means all adopted the reformed spelling of the name; and it would not be fair to attribute wholly to conservatism their disinclination to recog nize the innovation.
Is "Vergil" more American?
Not really, no, though the spelling has caught on more in America than it did in England.
Is "Vergil" more Latin-scholar?
Yes, definitely; I'd say it's more scholar, period, irrespective of whether they're of Latin or of poetry.
Is "Vergil" more 21st-century?
No, the restoration began in the 15th century, and has slowly moved forward in the scholarly world.
Was there a "schism" in the distant past? Or was there a period of Virgil-dominance followed by a Vergil-revival?
Less a schism, more of Virgil-dominance > Vergil-revival > coexistence