I just recently got this Chaffle cook book, and it says the following:
Egg-free Chaffles
"Swap the egg for 1 tablespoon (7 g / 0.3 oz) flax meal OR ground chia seeds, PLUS 3 tablespoons (45 ml) water. The texture will be crisper and less fluffy than chaffles made with eggs."
You could also try Egg White Chaffles
"Egg whites are ideal for creating a completely plain flavor with no eggy taste and a crispier result. To do that, swap the egg for 1 jumbo egg white OR 2 small egg whites to get a similar volume, or simply use 1 large egg white (slightly less volume)."