I've yet to visit but I've recently heard that the Benedictine nuns at Sant Angelo in Pontano weave their own cloth. They have rebuilt the original looms and will make all kinds of things to order - towels, sheets, Christening outfits, tablecloths, curtains, bomboniere.
Unlike most monastic orders, this monastery is growing and the average age is much lower than the norm aided by the younger women that are entering. Apparently, when they are officially accepted into the order, they arrive on their father's arm wearing a wedding dress and then leave in the nun's habit with their hair cut.
It's possible to go and visit the monastery to see the looms and the nuns at work. They will accept orders if there is something you especially want made.
Have a look at http://www.santamariadellerose.com/Arte-Telaio