Judging by the point in Act 3 when they read the letter that condemns them to their deaths and yet they do NOTHING to change that fate, I would have to say that they are existentially dead. Existentialism is about choices and choosing to act. Not choosing is not a choice -- it is to be existentially dead, and that is exactly when happens at the end of the play. They even ironically comment that "there must have been a point when we could have said no," but even then they don't go forward to action.
Throughout the play they comment on waiting for directions from the other characters. They WANT to act, but don't take much actions for themselves. Rosencrantz even comments that he "is only good in support." They ask who they are; they ask what their purpose is. They can ask all they want, but they should be doing something to create themselves and make meaning out of their lives -- it is not acceptable to leave all of the answers up to other people -- they need to take responsibility.
Ironically, we know that nothing can change for them -- their fate was written by Shakespeare over 400 years ago, and Stoppard can't play around with that too much.