How to make macarons without eggs?

Short answer, no. You can try other substitutes which will gel, but foaming requires a high concentration of protein chains which can act to trap and stabilize air bubbles. Folks make a lot out of the fact that these are classified as saponins, and equate them to soap. They aren't the same thing. Saponins just change the surface tension of water, and some of them taste bitter. But legumes are not the only foods you consume which contain them. Peanuts, spinach, asparagus, garlic, onions, oats, and others all contain them, just in lesser amounts. For all the talk about the potential hazard in the gut, there are also studies which suggest they are equally beneficial to the immune system- contributing to cancer resistance, as well as lowered cholesterol and glucose response levels.

In the case of your macarons, there isn't a substitute for the egg whites other than an aquafaba derived from one of the legume family that will work as well. Chia and flax will gel, but leave bits behind. Lecithin and agar agar are used to make culinary foams of liquids, but of course lecithin brings other dietary concerns. Plus you would still need a liquid to foam. I guess you could foam soy milk with lecithin, but that kind of kitchen chemistry always makes me nervous. Egg whites or aquafaba still seem to me the best choices. Unless you are eating macarons by the dozens on a daily basis, I don't really see a risk in either choice. I'd rather eat food that didn't require a degree in chemistry and a lab to create.

In the world of vegan egg replacements, your options usually fall into one of the below categories:
Fruit/vegetable puree
Seed gel
Blended tofu/nuts
Aquafaba

Fruit purees add moisture but don't bind very well. Seed gels are great for binding and add a little body. Blended tofu or nuts (usually cashews) are good for custardlike applications because of their silky textures. None of these can whip, so when it comes to whipped egg whites, aquafaba is pretty much your only substitute option. Coconut cream will whip, but because of its fat content it's more comparable to heavy cream, and so would melt if baked.
In any recipe, the amount of aquafaba consumed per serving is so small that I wouldn't worry about BPA or other compounds that it may contain.