Can having flat feet cause you to have a higher chance of MCL injury?

Possibly. With your arch down you may put higher stress on the inner side of your knee with walking and running. So it's possible but not the main reason most of the time.
Yes. Yes, at least theoretically. If you have severe flat feet you tend to have valgus ("knock kneed") ankles and secondary external rotation of your lower legs. This forces the knees to externally rotate and creates a valgus ("knock kneed") stress on the medial knee and medial collateral ligament (mcl).