Verizon doesn't lock their phones anymore (AT&T does). So you don't need to ask Verizon to unlock your phone - they'll have nothing to give you.
But, Verizon uses CDMA technology in the US for calls, and Europe uses GSM (like AT&T). Most modern Verizon smart phones can roam on GSM, but you may need to change a setting to do so - not sure how it works on an iPhone. After you get back from Europe and put your Verizon SIM card back in the phone, you may need to change it back to CDMA to work in the US again.
FYI, you can use Google Hangouts to make free calls back to the US with your Italian SIM card, even to US landlines, instead of using SIM credit for the calls. You can use Hangouts to call Italy too but it's not free (but pretty cheap - a few cents a minute, maybe more to call Italian cell phones). So if you really need to make only a few Italian calls and not receive any, and can stick to WiFi for that, you don't need to buy a SIM at all or use any Verizon roaming plan, but you'd have to buy a $10 USD block of credit on Google Hangouts to make calls to Italian numbers.