All of the previous posts are generally correct that the reason for there being so few Italian pirates in the Caribbean relative to the British, French and Dutch is because Italy (or Italian city states) had no colonies in the Caribbean. However, there were some Italian pirates/privateers especially in Spanish service during the period that are often overlooked. Italy did have a long maritime history and Italian sailors could often be found in Spanish navies. There were many Italian sailors aboard the Spanish Armada in 1588 for example, and obviously Italians ended up in the Caribbean as well.

The referenced Wiki page here of Italian pirates is definitely not comprehensive. The pirate historian Benerson Little writes this about the privateer Juan Corso (who was probably Spanish but could have been Italian and certainly had Italians in his crew including his first mate):

Little is known about Juan Corso, also known as Corzo Costa, Costa, and de Costa, other than as a sea rover. We don't know what he looked like, or when he was born, or his nationality for certain. His name may have been an assumed one. Corso was a fairly common Spanish and Italian surname, but it also meant "privateering," and so could have been a nickname. Other Spanish rovers--Blas Miguel Corso, for example--appended the word to their names.

There is a good chance that Corso was a Spaniard, but he could also have been Italian, Portuguese, Corsican, Sicilian, Slavic, Greek, or even a "Levanter," one of the many peoples of the eastern Mediterranean, all of whom the English referred to as "Greeks." His first mate, for example, was Giorgio Niccola from Venetia, which is either Venice or the region surrounding it. (Little, 107)

Juan Corso was a brutal privateer in Spanish service who ruthlessly hunted down English and French buccaneers during the early 1680s, eventually dying in a shipwreck off the coast of Florida in 1685. I'd highly recommend the book How History's Greatest Pirates Pillaged, Plundered and Got Away With It by Benerson Little which has a chapter dedicated to Juan Corso and his crew if you're interested in learning more.

Another Italian pirate/privateer in Spanish service who isn't mentioned on that Wiki list was called Matthew Luke and he is mentioned in the famous book A General History of the Robberies and Murders of the Most Notorious Pyrates published 1724:

About the same Time, a Guarda del Costa, of Porto Rico, commanded by one Matthew Luke, an Italian, took four English Vessels, and murdered all the Crews: He was taken by the Lanceston Man of War, in May 1722, and brought to Jamaica, where they were all but seven deservedly hanged. It is likely the Man of War might not have meddled with her, but that she blindly laid the Lanceston on board, thinking she had been a Merchant Ship, who thereupon catched a Tarter [an expression meaning to let an enemy board one's ship and then have the crew rush out from below decks to ambush them]. Afterwards, in rummaging, there was found a Cartridge of Powder made up with a Piece of an English Journal, belonging, I believe, to the Crean Snow; and upon Examination, at last, it was discovered, that they had taken this Vessel and murdered the Crew; and one of the Spaniards, when he come to die, confessed that he had killed twenty English Men with his own Hands. (44)

It should be emphasized again that these Italian privateers in Spanish service in the Caribbean weren't the same category as the British, French and Dutch buccaneers and pirates who they fought and hunted. I've never come across any historical reference to an Italian serving under any English, French or Dutch as actual buccaneers or pirates in the Caribbean as commonly thought of but that doesn't mean it didn't happen ever -- it was just extremely rare. The crew of the famous Welsh pirate Bartholomew Roberts, while almost entirely British with a few Dutchmen, had a single Greek called "Marcus Johnson" (name obviously anglicized like the Italian Matthew Luke) who came from Smyrna in Anatolia and he was hung as a pirate after being captured in 1722. An account of the trial says:

He says, he was born a Greck [Greek], and taken in a Dutch Company's Ship off Aura three Years ago, by Davis, the Predecessor of Roberts, that he would have made his Escape if he could have swimm'd, owns himself on board the King Solomon, and that he was put Centinel by the Quarter-master, as to the blue Shirt, he confesses it was shared to him out of a Dutch Ship they had taken, he also owns he had fired one Gun against the Swallow. Guilty. (40)

If a random Greek ended up on an English pirate ship then it's possible an Italian could have at some point. I've come across other more obscure nationalities as Caribbean pirates serving with the British and French, like Portuguese (Bartolomeo el Portuguese in the 1660s; even Moses Cohen Henriques, originally a Portuguese Jew whose family fled the Inquisition and served with the Dutch against Spain in the early 17th century) and Swedes (two Swedes joined the Scottish pirate John Gow in 1724; at least one Swede was with the English pirate Samuel Bellamy in 1717) and Danes (a number of Danes were with with the Irish pirate Edward England and tried to set themselves up as rulers over the natives on Madagascar in the 1720s) -- but their countries did all have colonies in the Americas and that's probably significant.

Lastly, although definitely fictional, an expanded edition published in 1726 of the previously mentioned General History of the Pyrates contains a fictional biography of a French pirate named Misson who supposedly set up a kingdom/republic in Madagascar during the 1690s (I've written a post about the real history of this here). Misson's sidekick in the story is a renegade Italian priest from Rome named Caraccioli who joins Misson in setting up his republic and expresses deistic, anti-religion, and egalitarian views probably inspired (or should we say the anonymous author of this fiction, speculated to be English journalists Daniel Defoe or Nathaniel Mist, was inspired) by the writings of Locke or the early writings of Voltaire. It is fiction but it was written right at the end of the "golden age of piracy" by a writer familiar with pirates and it's very fascinating.

Sources

How History's Greatest Pirates Pillaged, Plundered and Got Away With It by Benerson Little which has a chapter on Juan Corso and his crew

A General History of the Robberies and Murders of the Most Notorious Pyrates by Charles Johnson (an anonymous pseudonym), published 1724/1726

A Full and exact account, of the trial of all the pyrates, lately taken by Captain Ogle, on board the Swallow man of war, on the coast of Guinea, published 1723