Yes you can become physically sick when suffering from anxiety because when you're suffering from anxiety your body overproduces stress hormones.
When you're suffering from anxiety or other stress your body suppresses your immune system which makes it easier for your body to get sick since the immune system isn't able to fight off germs and other sickness.
I used to get really sick when I suffered from anxiety and the doctor did confirm that the stress and anxiety was a cause of me being physically ill.