Here are some options for switching out eggs. Each measurement is equal to 1 egg:
Leavening
1 1/2 tbsp of baking powder + 1 tbsp of warm water + 1 tbsp of apple cider vinegar
2 tsp of baking soda + 2 tbsp of warm water
1 tsp of baking powder + 1 tsp of vinegar
1/2 mashed banana + 1/4 tsp of baking powder
3 tbsp of ground flax seed + 1/8 tsp of baking powder + 3 tbsp of water (let stand for 10 mins)
1 tsp of yeast dissolved in 1/4 cup of warm water
Binding and moisture
1 tsp of soy flour + 1 tbsp of water
1/4 cup of applesauce or pureed fruit
1/4 cup of pumpkin puree or squash puree
1/4 cup of apricot or prune puree
1/2 mashed banana
3 tbsp of nut butter
1 tbsp of ground flax seed + 3 tbsp of water (let stand for 10 mins)
1/4 cup of silken tofu
2 tbsp of arrowroot + 1 tbsp of water
2 tbsp of corn starch + 1 tbsp of water
2 tbsp of potato starch + 1 tbsp of water
3 tbsp of vegan mayonnaise
3 tbsp of mashed banana
3 tbsp of mashed beans