Distract her when he's leaving so that she isn't thinking about it... You have to stimulate her before the tantrum takes control.
Hello,
there's not much worse than seeing a child you care for that upset. You may want to bring in your director to help counsel this family in how best to manage the separation compassionately. Her father's attempts at giving her extra comfort may indeed be making the situation worse, particularly if he is unwittingly communicating his own anxiety about separating.
As to the making herself throw up, its still a debate in the literature as to whether or not children can actually force themselves to vomit. Since we don't know for sure that they can do so, probably best to interpret it as a sign of severe anxiety, and not anything she can control.
Best Wishes
Rebecca Resnik