Should I delay adding barley when making vegetable barley soup?
It depends on how long the soup cooks for and whether you soak the barley or not. If the soup is going to cook for a couple of hours, you can just add unsoaked barley with everything else. Or you can soak the barley and add it towards the end of cooking.

If the soup is going to be cooked relatively quickly, you should soak the barley and add it with everything else. Soaked barley (soaked for a couple of hours) needs to be cooked for about 30 minutes.

It also depends on how soft you like the barley to be. Some people like it to near-disintegrate, some prefer it a little al dente.