During an at bat can a switch hitter change sides of the plate?

Yes, the player may surely switch sides; he may do so as long as the pitcher is not in the "ready position" (see rule 6.06),and, shall be called out if the batter attempts to switch sides during the pitcher's windup.(Incidentally, there has always been a popular myth in the baseball culture that a batter can only switch sides before there are two strikes. This is just a false statement). Read your baseball rulebook!

Yes, he can. It usually happens when the opposing team changes pitchers. Generally, left-handed hitters hit better against right-handed pitchers (and vice versa), so if a switch hitter is batting left-handed against a right-handed pitcher, and the other team changes to a lefty in the middle of the at-bat, the switch hitter will usually move to the right side of the plate to counter this.

There is no rule in the MLB rule book that states how many times a batter can switch sides during an at bat. However, once the pitcher steps on the rubber, whatever side the batter is on is the side he must bat from for that pitch.

Per above, there is no rule in the rulebook that states that he cannot change from one batter's box to the other in the middle of an at-bat. The only rule about switching boxes is 6.06b which says that he cannot switch boxes if the pitcher is in the ready position. Otherwise, no problem.

yeah they can i have done it before

addendum - PBUC 'Pat Venditte rule' Pitcher must indicate which arm he will use to throw the next pitch and then the batter must take either box. I am not sure what rule number in the PBUC that this falls under. If anyone has it please add it for me.