Are there any vegetarian-friendly egg substitutes that can be used in cakes?
Yes, there are multiple substitutes.
I can't explain it myself any better than here, but I have tried to summarize it:

The best substitute for egg in a dish depends on the dish itself (because the function of the egg isn't the same in every dish). As a general rule, the fewer eggs a recipe calls for, the easier they will be to substitute. Also consider how the substitute will affect the overall taste of the finished dish.

Possible substitutes:


Bananas and/or apple sauce (not flavourless, baked goods such as muffins, pancakes or yeast-free quick breads, especially banana bread; if you want something fluffy, make sure to add a bit of baking powder)
(Silken) Tofu (flavourless, best* substitute for quiche, fritatta or egg salad, can work in 'heavy baked goods' such as brownies or pancakes, doesn't work well in fluffy dishes; add bit of mustard, turmeric or nutritional yeast if you want the same colour as you would have used real eggs; I advice you to search for recipes that use tofu in stead of egg in the first place, as other recipe adaptations can be required)
Ener-G egg replacer (flavourless, works best* in baked goods, it can also act as a binder; if you chose to use a different brand, be careful to read the ingredient list, as some may contain egg white)


Less common substitutes (though I'm not sure if/how they work):


Flax seeds
Soda
Bread crumbs or oats (for savory dishes)


*) In the opinion of the site, I have no experience with it.