According to Bulbapedia:
After winning a battle, any Pokemon in the player's party that has this Ability and is not already holding an item has a 10% chance of creating a held item for itself, even if it did not participate in the battle. The Pokemon can find an item even if it is fainted.
So, each Zigzagoon in the party not holding a held item has a 10% chance to get an item on its own. Each Zigzagoon's pickup activates independently from the others.
In other words, yes, more than one Zigzagoon could get an item at the same time. After a battle, each of your Zigzagoon's pickups could activate, whether it be just one or all five. But because each pickup ability is seperate, I don't believe that if more than one pickup activated at once, then they would just automatically get the same items. The abilities would act independently from each other. There is a chance that if this happend they could get the same items, but this would be rare. For example, the chances of three of the Zigzagoon's abilities activating simultaneously AND all three picking up super potions are extremely small; it's more likely that each Zigzagoon would pickup a different item.
Also, any of the Zigzagoons could get an item first from pickup. It's not limited to the Zigzagoon at the head of your party getting an item first, then the second Zigzagoon getting the next item, then the third, etc. Any of the pickups can activate, so any Zigzagoon can get an item first.
Hope this answers your question!