While it is common to have at least 2 different spellings for Greek names, it is not so common with the names of ancient Romans. That's because the Greek alphabet is substantially different from ours whereas the Latin alphabet is substantially the same, so you wouldn't expect variable spelling for the name of Virgil/Vergil.
There are some differences between the letters of the alphabet that the Romans used and the ones used in English. The Romans had a few fewer letters. Consonantal "i" used alternatively for "j" and "u" used alternatively for "v" are potentially problematic. You might see Iulius or Julius, for example. But the Latin vowels and the English vowels are written in the same way. A Latin vocalic "i" is written as an "i" in English, and a Latin "e" is written as an English "e".
The Roman poet who wrote the great Latin epic The Aeneid was called Vergilius by the Romans. This is shortened in English to Vergil. Vergil is actually correct, but as in most matters of absolutes, there is good reason for the alternative.
According to Gilbert Highet in The Classical Tradition, the misspelling (Virgil) began early, possibly as the result of Vergil's nickname Parthenias which was based on the poet's sexual restraint. In the Middle Ages the name Virgil was thought to refer to his magical (as in the virga magic wand) powers.
It would seem that modern literature classes may spell Vergil's name Virgil. I never studied Vergil outside of the context of Latin, so for me, the name remains Vergil, but Virgil may by now be the more popular spelling.
I should throw in a reminder that Virgil / Vergil wrote the great Roman national epic, Aeneid, was heralded as a great poet even in his own time and has maintained his status among Roman writers, so if you haven't read Vergil (or Virgil), please do.