In Rome you have GLC straps only by appointment.
https://www.glcstraps.it/contatti/
But otherwise it seems to me the most well-known makers of straps in Italy nowadays don’t have bricks and mortars facilities,
In general I would suggest walking into watch stores and jewelery shops.
My most enjoyable strap shopping experiences in Europe were in neighborhood shops away from the touristy and affluent areas, where they sometimes had nice straps for good value.
But it really depends what type of straps you’re looking for, and for what type of watches.
In Milan there’s the thread below for vintage in Milan, also a jewelery and watch shop on the left side of the big plaza in front of the Duomo.
https://omegaforums.net/threads/vintage-watch-shops-in-milan.63944/
the makers below are well known but sell on the internet:
-for vintage style check out
@torrestraps on Instagram,
-more generally there’s the maker once sold through Hodinkee and Fratello, Jean-Paul Menicucci
https://www.jeanpaulmenicucci.com/contacts/
Finally if you really want walk-in strap shops, the ultimate shopping place would be Paris
😉