Most mug cakes will work without egg but will be a bit more crumbly. I use carton egg whites but as a vegan I used ground chia seeds blended with water to make an egg substitute. I have read you can make an egg substitute with cornstarch, a pinch of baking powder and some oil.

For oil I have used apple sauce from snack cups or baby food pureed pumpkin or sweet potato. The idea there is that the oil or oil substitute helps the cake retain moisture. I have also used instant potato flakes to make mug cakes fluffier and a bit moister. Sounds like sending oil is an option for you though.

I have used water in place of milk. It works but maybe isn’t as good. Half & half comes in small cartons and keeps longer than ordinary milk due to high fat content. I image full fat powdered milk or canned evaporated milk would work. Non-dairy creamer?

You should Google vegan mug cake recipes maybe. Example recipe: https://veganheaven.org/recipe/vegan-mug-cake-snickerdoodle-cake/

They use non-dairy milk usually though. Where I live you can find powdered coconut milk in the section of the supermarket or greengrocer’s with the Caribbean products. In the store it is cheaper than on Canadian Amazon where everything is far more expensive than on US Amazon. Coconut milk powder: https://www.amazon.ca/Grace-Coconut-Milk-Powder-Pack/dp/B00CPMJBWO/

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