The main issue is that first base is on the right side, so you can't apply the same shift to a righty as you would to a lefty. A real dramatic shift on a left handed hitter like Ryan Howard has the third baseman playing at about the shortstop position, the shortstop playing closer to the second baseman's usual position, the second baseman playing in right field, the right fielder playing in very deep right field, and the first baseman guarding the line. You can't do the same for a righty. On a right handed pure pull hitter, the third baseman may guard the line, the shortstop will move towards third, and the second baseman will be behind the bag. But the first baseman must be close enough to first to handle a throw. And the shortstop can't play in left field, because he'd be too far away from first base to throw the guy out. So you can't shift a righty nearly as dramatically as you can shift a lefty.
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I dont really know that there is a good reason why lefties don't play shortstop. One reason could be that, on balls hit in the hole between short and third, the lefty would wind up throwing across their body to first base meaning it would take an extra fraction of a second and the batter may be able to beat out the play and get on base.
They can be but it would be more difficult for them to make the throw to first in time because they would have to rotate their bodies completely while a right hander would just throw the ball from the position they field the ball in.
Lefties are most popular playing first but there are also some left handed third baseman in the MLB.