I haven’t found much lace in Rome but did come across several ecclesiastical shops selling vestments for priests, which had some lace in their windows. The selection here all came from one shop and included a variety of types. The main image shows needle lace and cutwork.
While the second image includes some handmade bobbin lace as well as chemical lace and the tape lace that is common in Brugges but made in the Far East. I assume they’re used to edge vestments and other church linen such as altar cloths.