Yes! You will need to choose vegan bread, then use non-dairy milk and an egg replacement. I prefer making it with a classic egg substitute such as Just Egg or a mix like the ones from Bob's Red Mill. This may require you to whisk it up with water before mixing it into the dairy milk you prefer. While I have not tried it, you could also use a flax egg or even add in chia seeds to thicken and help create the texture you want in the bread pudding.