Ciao Robert, welcome to our forum!
Well you'd think there should be shops that do this, but to be honest the idea of suit hire/rentals here in Italy still has to take off.
I found an article specifically on a shop in Siena that does do rentals for wedding dresses and they say it isn't a common practice here, but it seems they specialize, in particular, in dresses and not in suits for men.
In any case you might want to contact all the ones I found and ask - they might be open to the idea, even if they don't generally offer it as a service.
Sappiamo l'importanza che ha per te il tuo matrimonio e vogliamo aiutarti a organizzarlo per fare in modo che sia il giorno più felice della tua vita.
www.matrimonio.com
An alternative would be to consider, of course, to buy yourself a dapper Italian suit and then have that as a souvenir to use back home on all occasions that need it!
This shop L'Atelier dei Sogni -
https://www.latelierdeisogni.com/who_we_are - has a selection section for men, but not sure they offer rentals.
One that does offer rentals for men is in Florence, do you happen to pass through here on your way to the wedding?
Here's the page for you to contact them, with already some ideas of what you want:
https://www.sognodisposafirenze.it/noleggio/
Then if none of the above work out, there is definitely this one in northern Italy that offers the service AND will mail your suit to you. If you have pretty standard size you wear (remember to find out what your size is in Italian sizes), then this could be an option!
Noleggio Abiti Cerimonia Sposo Smoking Tight Milano. Vasta gamma di Abiti Noleggio ed Ex Noleggio. Noleggia e Affitta lo Smoking a Milano a sole 95€!
www.ilmiosmoking.com
Hope I've offered some places to start and that might work out for you!
