After speaking to Piola's manager, they now offer vegan cheese for their pizzas. It's Daiya brand, so it's soy-free, gluten-free, and dairy-free.
The minestrone on the new menu is vegetarian (no beef or chicken broth) and there's a note about the vegan cheese right on the menu!
The gnocchi recipe is no longer vegan, which makes me sad, but now that I have Daiya, I can have pizza, pasta and any salad with it at no extra cost, so I'm pretty satisfied.
Still no soy milk option for your post-dinner cappuccino, but I've taken to having espresso like a real Italian and not the manga cake I was raised as. Would love it if they offered a dairy-free dessert as well.