I asked a similar question about the Royal Guard Champion in a different mission with very similar mission rules (one action after each hero, instead of a "normal activation"). FFG's official response was:
When a figure activates in this capacity, each of those actions contains its own separate “start” and “end” of activation and resets the activation count. So, Jyn can Quick Draw the RGC when it takes one of those actions and it may use Brutality as often as it wishes using those actions.
So Jyn can use Quick Draw on Vader in that context, since each action counts as its own activation. Unfortunately for Vader, he can't use Brutality four times per round the way RGC can in the other mission, because Vader's rule says "move 2 spaces and make an attack" which forbids special actions such as Brutality or Force Choke.