This is the original sentence: "I came to Italy last year for a business trip."
can I say: "I came to Italy last year for a vacation." ?
I want to emphasize "not a business trip".
Either for vacation or on vacation would sound good to me (rather than for a vacation). Also note that came is only appropriate if it is already known to your audience that you were (or still are) in Italy. Otherwise use went.
I went to Italy last year for vacation.
I went to Italy last year on vacation.