Italian supermarkets stock basic pet food brands, which covers most casual restocking needs on a short trip. For a specific brand, a larger selection, or a prescription diet, a dedicated 'negozio per animali' (pet shop) is the better bet — major chains like Arcaplanet and Maxi Zoo are widely available in cities and larger towns, stocking everything from everyday food brands to travel accessories and health basics.
Worth packing enough food for the trip if a pet is on a specific brand or prescription diet, since large chains carry popular international brands but not every niche one.