Can you veganize this recipe?

I too miss challah, but I doubt that I'd have much success making it both vegan AND gluten-free. However, on the egg front, I'd suggest a relatively new vegan product, Just Egg, which comes in liquid form ready to use in cooking and baking. Every recipe I've tried it in so far has come out well. I'd expect it to be especially effective in making challah because it's yellow; I recall the sunny color of challah being part of its attraction.

Would another dairy-free milk substitute work instead of oat milk? Almond milk or something else?

Yes! You can use aquafaba for the eggs. 3 tbs aquafaba equal 1 egg. I've tried it and works well!

Here’s a recipe that uses the liquid from canned cannellini beans in place of the eggs. https://food52.com/recipes/85079-vegan-challah

There's already a suggestion of oat milk for dairy milk.
For the 3 eggs in the dough, use pureed soft tofu (used in baking cakes and breads for years when I was a vegetarian).
For a vegan replacement for the egg wash, see heart of a baker site for suggestions or just omit that step.