For most shooters standing is the most unstable position so their accuracy is very much reduced. Are they still able to hit their target? For the most part yes, but they won't be quite as consistent and their grouping will be much less tight. Most shooters will minimize sway while standing by keeping their shooting platform tight but this changes when you introduce movement.
Naturally the movement caused by walking or even running will cause your weapon to sway additionally. You also lose the benefits of a tight shooting position and begin to rely primarily on your upper body strength to support your weapon. This will cause your weapon to sway even further.
You also have to consider the weapon your using. A pistol will obviously be lighter and more easily controllable than a rifle. However, pistol are for the most part less accurate than your standard rifle. If you a rifle, something along the lines of an M4, you have more wight and more bulk to contend with. All things considered, could you hit your target while moving? At 20 meters or so, at a motionless human sized target, probably. But that is an ideal situation. Should expect to hit a moving target at 50 yards, in the wind while your jogging? No, you probably shouldn't.