We've never had trouble getting a taxi to any Italian airport for a 6:00am or 6:30am flight, but we have had trouble with the hotel-to-airport shuttles that early.
In Bari years ago, we stayed at the airport hotel Hotel Parco dei Principi intending to use their shuttle. But we actually ended up walking to the airport rather than wait for a taxi when the shuttle wasn't going to depart early enough for some reason. Not ideal, there weren't sidewalks, the sun wasn't up, but it was a short walk.
We'll probably be making the same decision this fall for a 6:30am flight out of Venice airport: choosing a hotel within walking distance of the airport just in case.
If I were you, I'd probably book the taxi for earlier than necessary and have the number of another taxi handy, just in case. And check in online. A good hotel, B&B, or apartment host should help you with the numbers to call or call for you.