Naxos is a very veg-friendly island, especially inner city. My favorite place was Aktaion, a pastry which serves vegan verrines and coconut ice-cream.
For a vegan breakfast, Anna’s Organic Gardencafé is a lovely option. You can also purchase extra organic stuff (from food to cosmetics) into their shop.
When it comes to junk food, Yasouvlaki is the best as they serves French fries, wraps and falafels with a vegan mayo.
Pizzadelia is all about Italian food with a bunch of veggie options. And if one of them is not good enough for you, you can custom your own gluten-free piece !
It’s pretty easy to find nice vegan options everywhere, way easier than anywhere else in the Cyclades ! I even found a bucher who sells vegan cheese, nuts and legumes !