Yes, you can run with flat feet and it poses no major difficulty in the initial run. However, when you are running, the arch of your feet gives you a natural springboard-like movement that helps you accelerate and which also helps you take off some pressure from the toes and the heel. When you are flat-footed, however, you lose this advantage. A short run might not feel difficult or different at all, but it is when you have to go a longer distance than you might face some problems. Wearing a good set of cross trainer shoes can fix this by providing you artificial arch support.