You can leave out the eggs, or replace them with tofu if you want some texture. Replace the chicken stock suggested for a meatless (!) version in the recipe you link to with vegetable stock.
If replacing the chicken stock changes the texture of the soup too much, you could add some (vegan) gelatin or similar gelling agent. However, the potato starch n the original recipe is the more effective thickener, so you might not need to add gelatin.