Making potato pancakes without egg
The non-vegan ingredients are the egg and the ghee (clarified butter, browned to develop the nutty flavours).

For the ghee you can substitute the same quantity of olive oil, possibly with some loss of flavour.

The egg is harder to replace. You could try a commercial egg replacement: these are powdered starch (one product uses potato and tapioca flour, so is gluten free) with assorted emulsifiers. A substitute widely recommend on the web for savoury pancakes is a gel made from flax or chia seeds: mix 1 tablespoon of seeds with 2.5 tablespoons of water and leave for 10 minutes for the seeds to fully hydrate.