It is easy to confirm that the soup is vegetarian. If it contains cheese, you need to specify that the cheese is one of the approved type cheese on the second sheet of the year-round kosher list and that the Renet is not from animals. In any restaurant, supermarket, chain, or large store, you can rely on the waiter. If he’s not sure, then have him ask the chef. The reasons they can be trusted is for several reasons, including that he has no reason to lie, they’re not paying him enough, and the risks of allergy lawsuits are too high..
Once you confirm that the ingredients are permissible, your biggest issue then is Bishul Akum (cooking of a non-Jew). If a Jew owns the place or there’s a Jewish chef, even non-observant, it’s fine. Bishul Akum literally refers to the cooking of a non-Jew. That does not apply to Jews, even non-observant.
Otherwise, you need to understand the following. All puréed soups are fine in regards to bishul akum, because even if there are some items within it that is in itself a bishul akum issue, since the majority is water, the other stuff is batel. This bitul works if 51% is water and/or other non-bishul akum items.
If you have something that’s be’en (not mixed in well) for example full barley or cut potatoes etc, then you cannot eat those items.
Bishul Akum only applies as a prohibition to items that cannot be eaten raw. Potatoes, barley, rice, egg, and so on are a problem. Carrots, broccoli, cauliflower, chick peas, peas, mushrooms, and so on are not a problem.
Feel free to post here or in a new post any soup examples and we will analyze them together.