I'm a vegan of 4.5+ years.
I've looked online many a time for vegan alternatives to recipes. Some are good. Some are shit.
I've learned that if you still want to make good food as a vegan, sometimes that involves learning substitution "tricks" that can be used to turn a non-vegan dish into a vegan one.
For eggs, I've mostly heard that you can replace those with soaked chia or flax seeds. You use the pectin that dissolves out of the seeds as the binder.
I have also seen people only claim that apply sauce or bananas are good vegan substitutes for egg. I haven't really tried that out before in it's own, but I imagine that the combo of ( soaked chia/flax seed + banana/apple sauce ) is what emulates eggs the most. You get the binding power from the seeds, with the wet nature from the banana or apply sauce.
FYI, vegans look for ways to "veganize" dishes all the time, especially at restaurants. Seldom do I want to take my friends or family to vegan-only restaurants because they tend to have "fat vegan" food. I'm better off enjoying a meal with them at a vegetarian/omnivore restaurant, which may or may not have vegan options. You have to learn to make substitutes on the fly to maintain your practice.
Just my 2¢