Matt, not according to the proposed errata / FAQ - page 6:

Question:
When vanquished do I continue to track Stamina and or Wound loss past 0 and into negative
numbers?

Answer:
No, you stop at 0 stamina/wounds


This is in line with the first sentence under the Vanquished description on p. 316 of the A:RPG:

"Characters that are Vanquished rarely die unless their entire group is Vanquished at the hands of a murderous enemy."

I believe the exception to that most commonly encountered is the spell "Strength of Fallen Foes".

With a sweep of his hat,

Paul

Okay, say you are vanquished by wounds (0 wounds remaining) and you make your stabilization roll.

If an enemy THEN goes over to you and sticks a dagger through your throat, I as GM would say that that enemy is DEAD. You are helpless, and can be coup de grace'd. If you are vanquished by STAMINA and not wounds and they do this, I would say it does a Wound damage because 1) you have no remaining Stamina, and 2) are considered helpless. Stamina is merely your ability to continue fighting before exhaustion, so you still have all your (remaining) wounds to be taken when you fall from that method of damage.

One would need to make another roll to avoid death (with all penalties) but it is not automatic.

You still have an Avoidance of 11 to hit (they can fail to coup-de-gras).