Yes, you can make bean/lentil/chickpea/millet burgers and veg fritters without egg (and I never bother putting egg in meat burgers either). Breadcrumbs or flour are good for binding, or gram [chickpea] flour if you're wheat-avoiding. Plus some olive oil or just water if the mixture needs more stickiness as opposed to more density. It is all a bit trial and error I'm afraid, but you'll get there!