Yes, social anxiety runs in families. Social anxiety disorder, also known as social phobia, is inherited from family members, and also has environmental influences. Environmental influences which increase the risk for social anxiety include psychopathology in parents, overcontrolling parents, adverse life events (abuse, relationship problems), different cultural values, and low income (Carandang, 2014).
In summary, social anxiety runs in families, with both genetic and environmental influences.
Reference:
“Chapter 4 – Social Phobia.” Anxiety Protocol. Carandang C. 2014. Healthy Mind Research Corporation.