Chia eggs and flax eggs are a staple in egg-free baking.
Good news, though, they're SUPER easy to make.
For one "egg", simply mix together 1 tbsp chia seed or ground flax* and 3 tbsp water.
You should let that mix sit for about 5 minutes so that it becomes thick, but I'd be lying if I said I waited every time, and I've yet to have a baking disaster (at least not related to my vegan eggs...), so let's say the 5 minutes are optional.
*Please note, your flax MUST be ground. Flax is grown as a seed, but it needs to be ground for it to work in this case. You can purchase either "ground flax", or "flax meal", but if you find that flax seeds are cheaper, simply toss them in a coffee grinder and grind them on your own. Easy peasy!