See below. The vast majority of flat feet are inherited and usually traced back to mom and dad. Some are post traumatic but on rare occasions. Apple doesn't fall far from the tree when it comes to flat feet.
The only things. I would add to dr. Cerniglia's excellent answer is the gene for flat feet displays "incomplete penetrance, " which means it can skip one or more generations. Also, there are 2 kinds of flatfoot deformities: flexible flatfoot, far more common and easily treated, and peroneal spastic flatfoot, which is a fairly disabling problem. See a podiatrist.