Aquafaba (the liquid in cans of chickpeas or other white beans) is a really trendy, but apparently effective substitute right now. Food52 has a bunch of articles on it, including this one: https://food52.com/blog/13271-a-magical-new-egg-replacement-that-s-already-in-your-pantry. Good luck!
Another option (than the flax fake egg already suggested) - I have successfully used soft tofu to replace eggs in baked goods.
Use same volume as eggs required by original recipe (so 3.25 to 4 tbsp per American-sized large egg...see linked chart).
Buzz the tofu in a food processor or with a hand blender.
Then add to recipe.
This works for both sweet goods raised with eggs and baking powder and yeast breads or cakes.
No tofu or soy taste discernible in finished products.
The SOFT tofu is important, as other textures (regular, hard, etc) will not puree nor disappear into the batter or dough.
Egg size chart:
https://en.wikipedia.org/wiki/Chicken_egg_sizes
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Use same volume as eggs required by original recipe (so 3.25 to 4 tbsp per American-sized large egg...see linked chart).
Buzz the tofu in a food processor or with a hand blender.
Then add to recipe.
This works for both sweet goods raised with eggs and baking powder and yeast breads or cakes.
No tofu or soy taste discernible in finished products.
The SOFT tofu is important, as other textures (regular, hard, etc) will not puree nor disappear into the batter or dough.
Egg size chart:
https://en.wikipedia.org/wiki/Chicken_egg_sizes