How could this recipe be adapted for vegans (no butter, milk, or egg)?

The egg whites serve as a binding agent here, so I'd probably just use 3 or 4 tablespoons of flour (gluten free mix or regular) instead, and Miyoko vegan butter, and sweetened condensed coconut milk (Whole Foods Market sells it in the baking aisle). I've used sweetened condensed milk in this recipe once before, but found the macaroons much too sweet, so I'd use unsweetened coconut instead of the sweetened called for in the recipe. I might add that the egg does make the macaroons a bit "eggier" than most - thus the "custard" in the name. The custard element did not, however, come through as much as I expected. Instead of using flour, you could try a vegan egg substitute, following the substitution guidelines on the package. ;o)

Well, you could use aquafaba for the egg, margarine for the butter, and sweeten and condense a nut milk, but it would probably be better for you to find an already-tested vegan macaroon recipe.