Yes, social anxiety disorder, also known as social phobia, can lead to depression. With social anxiety, you avoid social situations, to the point where you are isolating yourself to your home. This can lead to bad outcomes with your job, relationships, and community work. With the social anxiety, you first start to think that people will make fun of you, or you will embarrass yourself in front of people. So you feel anxious around crowds of people, and it severely affects your life, as you would rather just stay at home, hiding.
Soon, you start thinking that all is hopeless and you are useless. This type of thinking can then lead to feelings of depression, and you try to reduce your depression by withdrawing and isolating yourself further. But these isolative behaviors only serve to maintain the negative, vicious cycle of depression.