Why do The Sopranos leave off the last vowel in Italian words?

This actually happens in Italian itself, particularly but not exclusively in the dialects spoken south of Rome, including in Sicily where such folks originate. (But see the Mark’s article below for some nuance there.)

Unlike northern Italian, which has a syllable-timed rhythm to it and so vowel reduction does not occur, southern Italian tends more toward the stress-timed end of the spectrum. This means that its vowels which are not stressed are shortened and reduced, and may even (seem to) be dropped completely. See this Language Log posting on “Italian: To vowel or not to vowel” by Mark Liberman for more.

The thing is, this is more common in Italian than English speakers may realize. To see that it happens pretty normally, just think of the familiar phrase il dolce far niente, where far is a shortened form of the normal verb fare. A less familiar phrase is a fior d’acqua, where fior is a similarly shortened version of fiore.