Debatable

At least Italian citizens would be bound by European Union laws. While I am not certain that it is already the case for Italy, there is a Council guideline telling all countries to apply extraterritorial laws for AOC, so in theory all countries should have it or start introducing such. Then, some countries consider crossing international borders and having sex as trafficking anyway, much like US law considers state borders - again, I am not certain this is the case with Italy, and I am actually inclined to think it is not - but anyone in doubt should check.

So, in general, if you are a EU citizen (or a small bunch of other countries), I wouldn't recommend you using the Vatican to break AOC laws, at least not without being sure neither extraterritoriality nor trafficking laws will apply. For US citizens the former certainly does.