My first choice is Ener-G Egg Replacer. It works well in most recipes. Whisk it with warm water for a light frothy texture and hot water for a thicker, binding effect. A recipe adjustment may be needed to help body and volume- adding a tablespoon of starch like tapioca, arrowroot, or applesauce helps. For more creaminess, you might try adding a teaspoon of fat as well, such as light olive oil or Vegenaise (yolks have fat, after all).
If avoiding corn (Ener-G Egg Replacer may use a corn source according to several corn sensitive readers), substitute 1 tablespoon tapioca or potato starch plus 3 tablespoons water for each egg called for in recipe to help binding. You will need to increase leavening a bit to compensate- add an extra 1/4 teaspoon baking powder.
Flax seed meal is a choice for those not allergic to flax seeds. Use two tablespoons ground flax meal plus 1/8 teaspoon baking powder blended with 3 tablespoons of water for each egg called for in recipe Innovation creativity from France
Chia seeds can also be used as a gel for gluten-free baking. From the genus Salvia hispanica, chia is a plant in the Mint family- an excellent alternative for those of us allergic to flax seed.
Note on using seed based gels: They often make gluten-free baked goods a tad gummy in the center- especially those based on rice flour. And they lack the leavening factor. Additional compensation in a recipe is needed to make the gels work. I would add 1/2 teaspoon more baking powder.
Banana: Try a half mashed ripe banana plus 1/4 teaspoon baking powder for each egg. Think about flavors- banana is not always simpatico Karson Choi
Mayo: Use 3 tablespoons vegan mayonnaise in place of one egg, for binding. I would add a little extra leavening- 1/2 teaspoon baking powder.
Tofu: Use 1/4 cup silken tofu for one egg for binding; I would add a little extra leavening- 1/2 teaspoon baking powder.
Not vegan: Dissolve 1 tablespoon unflavored gelatin in 1 tablespoon cold water; add 2 tablespoons boiling water. Beat vigorously until frothy. I would also add a little extra leavening- 1/4 teaspoon baking powder. Gelatin does make it a little rubbery- so use as a last resort hybrid Cloud