Maybe - the DM decides the order goblins act in

Creatures acting on the same initiative just means that the creatures all act one after the other. Otherwise, there is no difference in the turns and what they can do with them compared with any other creatures. That means each goblin gets an action and movement. If they are able to act, they can use that action to shake another creature awake if they wish.

The DM will have to decide the order in which the creatures get to act (because they are a tied group of identical creatures that all act at the same time), but if the unaffected one end up going first he can indeed start a chain that wakes them all up.

However, do note that this implies that they are all grouped very tightly together with no enemies interfering with their movement or actions (opportunity attacks and readied actions come to mind). Additionally, they are all sacrificing their actions for the round, thus giving the PCs a big boost in action economy for the round.