Egg substitue in enriched breads
I use ground flaxseed and water, which is a common egg substitute - I prefer the flavor of the "golden" or yellow seeds .vs. the brown ones (and unlike the brown seed grower's propaganda, I do taste a difference) but either will work for the egg substitution - chia seed is evidently another common substitute.

I consider it unsafe (or unwise) to assume that the only function of the eggs is to "enrich" - I find that the "same" bread made with or without eggs behaves quite differently in terms of how it rises, so I would suggest sticking to a substitute that gives you more complete functional replacement, rather than, say, "add some oil."

I'm not myself a vegan but I have baked things with keeping vegans happy in mind. As such I have worked with eggs and with ground flaxseed and water, and so far I consider the flaxseed mixture as good of an egg substitute as I have used, though still not the same as (nor indistinguishable from) actual eggs.