I am stationed in Germany near Ramstein. My affiliation with the military here is as a .civ.mil.
It depends on the SOFA with Italy. There are two people from my organization that are in Vincenza. He hunted at Ft Drum and if I remember correctly he looked into hunting in Italy but I don't know if he got anywhere with it.
Here in Germany the SOFA allows us the opportunitty to obtain personal firearms and import them AFTER we have passed the test for a hunting license or a sports shooter license. Sports shooting license requires an additional 12 months of 'training' on what you intend to "sport shoot" before you can get a weapons registration card. With a hunting license I can buy any amount of long guns I can legally store. I am allowed two pistols. The hunting course was 15-20 hours a week for four months. Then I had to get a safe that met requirments. Then I had to buy a rifle in order to get my weapons card. As strict as they sound I can walk into a store and take a rifle home without a background check. The hunting license is my permit to purchase. I have two weeks to register it. Pistols require a permission to purchase stamp by caliber. Then I can get one.
I reload here and I took a two day class in order to get a powder license (pulverschein). I had to have a hunting license to take that class.
I even have an awesome range 2-3 miles from my house I can shoot at all day for 5 Euros.
As difficult as this all sounds it wasn't that bad. It was a little costly like starting over. There was timing on paperwork, steps to follow, etc. But the biggest PIA I came across was all the little things. I didnt think I would have the opportunitty to shoot or hunt so I left it in the US. For example I had a Dewey cleaning rod for .30 back in Colorado. After I got here and got set up I wished I had it...Went looking and found them for about 100 euros. Want a sling for a rifle? Decent ones start at 40-50 Euros. I didnt like the prices so I get on Amazon and have shipped what I can...More and more they wont ship. My point is if you have a Caldwell rest you like? Bring it. Like all that reloading gear? Bring it. Do you have cleaining kit you like? Bring it. My point is that If you think there is an opportunitty then bring everything you think you might need so you dont have to buy it again.
All the Rod & Gun clubs have been replaced by "MWR Outdoor Recreation" The website in italy is
https://italy.armymwr.com/programs/outdoor-recreation . I would get in contact with them for the straight answer. Outdoor Rec here in Germany has classes for hunting licenses and sports shooting in Baumholder, Kaiserslautern, and Grafenwoehr.
If you want to look at an English Website that sells to hunters and shooters check out Akah.
https://www.akah.eu/ . Want to look at ebay for guns?
www.egun.de is where you go. Type in "M1A" "Garand" "Luger" "M4 Carbine". Yep I can buy those also. No magazine restrictions unless Im hunting.
If you have any questions PM Me.
Edit: I forgot to mention to get customs to stamp off on brass, bullets, and any animals trophies before you leave. Legally you are required to fill out paperwork to reimport them.