Shadows of the Empire- Action : Place a target enemy objective at the bottom of its owner’s objective deck.

Prince Xizor- Reaction : After an objective leaves play, capture 1 card at random from each opponent’s hand

My question is, and I apologize if this has been answered and I overlooked it, if I play Shadows of the Empire 2 times in one turn, does Xizor's affect trigger both times, or does his reaction just hit one time per turn?

If it was only once per turn it would say so.

By default, Reactions and Interrupts can be used once per triggering condition as often as that trigger occurs (in Xizor's case, as often as an objective leaves play). Some cards specifically limit the ability to once per some time period (such as Darth Vader's Reaction being limited to once per turn). Otherwise, the only limit is the number of times the trigger occurs.