Vulnerability has no special effect on death saving throws
The rule for taking damage at 0 HP is:
Damage at 0 Hit Points. If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead. If the damage equals or exceeds your hit point maximum, you suffer instant death.
The two failed throws comes only from a critical hit not from other forms of increased damage.
Critical hits and vulnerability are two separate mechanics with two separate names and that function differently, and there is no reason to conflate the two.1 If the death saving throw rules were meant to include them, they would have listed vulnerability by name in the description.
Of course, because the damage to a vulnerable creature is increased they do have a greater chance of suffering instant death, but they would only take one failed death saving throw assuming they survive.
1 - Vulnerability actually doubles damage, but critical hits only has you roll the damage dice twice. It is a subtle but important distinction, but critical hits do not double the damage.