For vegans the menu is limited, but that doesn't matter. Get the Marinara (the original - some would say - Italian pizza, simply made with sauce and garlic) for 12zł and you will love it. I promise. I've ordered it at least 5 times and it has been consistently great. Their dough is thin and chewy and the sauce is tangy and perfect.
Piec Na Szewskiej is often busy so sometimes you might have to wait a while for your pizza. I'm not a fan of waiting, but in this case it's worth it.
If I remember correctly the menu is only in Polish (they might have an English menu, but haven't asked) so here is the guide:
2 green stars means the item is vegan.
1 green star means the item is vegetarian.
Cons: sometimes it's a long wait