vegan version? is there a way to make this without the egg white?

Try aquafaba, the egg white is mostly just to get everything to glom together.

After a quick look, found a vegan version of nuts made without egg white.
Not exactly the same as the one you posted, but could be adapted.
https://www.crowdedkitchen.com/vegan-sweet-and-spicy-nuts/#tasty-recipes-7532-jump-target