I need an egg sustitute for French Apple Cake by Dory G. It also uses bkg. powder. Would flax eggs work?

Can you link to the recipe? Because how many eggs are you replacing? If it's more than two or three, maybe consider commercial egg replacer? Depending on how dense the cake is anyway, apple sauce is also a great sub - one container per two eggs.

If by "flax eggs" you mean ground flax seeds and water, probably yes.
Or silken tofu, same liquid volume as eggs in original recipe, pureed and mixed in.