How do I make a pancake without egg?
There are multiple recipes for eggless pancakes online.  Most increase the amount of baking powder to help with the rise. Many use milk and butter, so they are not vegan.  If you are looking for a vegan recipe, those exist too.  They also tend to increase the baking powder, and often include non-dairy milk.  Whether or not they will sufficiently mimic the fluffiness of a traditional recipe is going to be completely up to you.