Non-dairy yogurt usually consists of a base of either soy, nut, rice or coconut that is fermented with lacobacillus cultures. Depending on the base and fermentation method, hydrocolloids such as tapioca starch, agar or xanthan gum may be added to increase viscosity.
Non-dairy yogurt isn't actually an egg replacer that's on par with eggs. It's about as effective as traditional dairy yogurt as an egg replacer in baking applications. This isn't much but it works great in things where significant binding power isn't needed such as vegan cakes, waffles and pancakes. It won't help you get things very dense and chewy like traditional eggs can but it's great for holding in moisture and adding flavor complexity at the same time.
One egg contains about ¼ cup of liquid so using non-dairy yogurt as a substitute in a recipe consists of:
¼ cup non-dairy yogurt