Among Four Swords manga fans, there are two ways that we see Vio's name pronounced: "vee-oh" and "vye-oh". Which pronunciation is the "correct" one is debated.
Most people will stick with whichever one they personally use regardless of what is "correct". However, for the purposes of analysis of the original Japanese version of the manga, I want to offer what I believe is the "correct" pronunciation of the name.
In Japanese, the English word "violet" is sometimes transliterated as ヴァイオレット (pronounced vye-oh-reh-toh). Red says this word when he first tries to think of a name for Vio. The furigana next to the kanji character for the color purple, 紫, is written as ヴァイオレット to show how he pronounces the word aloud (see the photo below).
However, when he shortens the name to the familiar "Vio", he says, 「ヴィオでいいかな。」 (Again, see the photo below.)
The word ヴィオ is pronouced "vee-oh". This is different from ヴァイオ, which is taken directly from ヴァイオレット and pronounced "vye-oh". In other words, Red deliberately chose the nickname to have a special pronunciation.
Personally, I have always pronounced Vio's name as "vye-oh", but I believe that "vee-oh" is, objectively, the "correct" way to say the name - or at least, that is how the authors intended him to be called. It doesn't really matter how you pronounce the name as long as people know who you're talking about.