It is quite simple to order. Just install the Domino's app on your phone and then when ordering a pizza simply scroll down all the way until you find the "vegan menu." Alternatively you can simply select a vegetarian pizza and then press a button that allows you to replace the dairy cheese with vegan cheese.
According to the video above, the vegan cheese used comes from Follow Your Heart. The key ingredient in these vegan cheeses are coconut oil, which is interesting because many vegan dairy products nowadays seem to use coconut, e.g. coconut yoghurt.