Eggs are typically used in a recipe as a binder. There are many substitutions available. You will still need to check that there are not other ingredients to avoid. But, no eggs, no problem! We've gotcha covered.
Some substitutions are:
Baking powder & Oil - for each egg use 2 TBS water, 1 TBS oil & 1/2 tsp baking powder (works well for replacing 1-2 eggs).
Powdered commercial egg replacer – Read label, some contain egg, potato starch, tapioca flour, leavening or corn.
Arrowroot powder – see post on Arrowroot Allergy
Psyllium seed husk powder
Tofu – made from soy
Tapioca flour - LOVE this gel from Cake & Commerce
Banana – adds distinctive taste. Watch if you have OAS or allergy to ragweed/latex.