Energ-G Egg Replacer
From a general perspective, it might be hard for people to believe that vegans can still eat a wide variety of foods while maintaining a diet that restricts meat, eggs, and dairy. However, when you've got a better understanding of what makes a suitable substitute for what in any given recipe, you can usually fare pretty well when recreating standard dishes in a plant-based way.
One of the handiest substitutes you can have in your kitchen is the Ener-G Egg Replacer. This product is extremely common in vegan recipes (especially baking) and is a simple mix of tapioca flour, potato starch, and a leavening agent. This is important because the tapioca works as a binding agent, the potato starch works to create texture, and the cream of tartar-based leavening agent allows baked treats like muffins and quickbreads to maintain their ability to rise well even without regular eggs.
One 16-ounce box of Ener-G Egg Replacer is around $7 each, which might seem a little costly, but it stores well and should last you quite some time. To substitute, 1 egg in any recipe is equivalent to 1 1/2 tsp dry Egg Replacer plus 2 tbsp water, so you're only ever using a little bit. You also should be able to find this in the baking section of your supermarket, but if not, one quick trip to Whole Foods or another local natural grocery store should suffice!