Does the vibration due to the earthquakes distort the interference pattern of LIGO's interferometer?
Yes and no. The measurement systems are isolated from ground vibrations by a complex system of supports, but earthquakes will still affect them. There are passive systems which try and isolate the mirrors and active ones which sense local ground movements and cancel them out.
The main defense is the second LIGO in another state which won't feel the same earthquake at the same time. A signal is only likely to be astronomical if it is detected by both of them at (almost) the same time.