This musket has two parallel barrels; each barrel can be shot independently as separate attacks, or both can be fired at once as a standard action (the attack action).
If both barrels are fired at once, they must both target the same creature or object, and the gun becomes wildly inaccurate, taking a –4 penalty on each shot. Each barrel of a double-barreled musket uses either a bullet and a single dose of black powder or an alchemical cartridge as ammunition.
By the first quoted paragraph, both shots can be fired as an attack action. Hence, vital strike applies:
When you use the attack action, you can make one attack at your highest base attack bonus that deals additional damage. Roll the weapon’s damage dice for the attack twice and add the results together before adding bonuses from Strength, weapon abilities (such as flaming), precision-based damage, and other damage bonuses. These extra weapon damage dice are not multiplied on a critical hit, but are added to the total.
But vital strike only allows one attack that deals additional damage.
Hence, the benefit of the vital strike only applies to one of the double-barreled attacks.
An alternative reading of vital strike is that it allows only one attack, and that only allowed attack is with bonuses. I don't see any way of deciding which interpretation is more correct.
To be more explicit, the different readings are:
- When using the attack action, vital strike allows you to make one of the attacks you make (at highest BAB) to have additional damage.
- When using the attack action and when making only a single attack, vital strike allows you to make that attack (at highest BAB) with extra damage.
Vital Strike was not written with double-barreled weapons (weapons that allow two attacks with a single attack action) in mind. Despite my best efforts to search the Paizo forums and FAQs, I did not find a clear developer clarification on how these things interact. As a result, the most basic answer to this question is that the ruling is up to your DM.
That said, let me take a stab at making a ruling:
By strict "rules as written", double-barreled weapons modify the attack action, causing it to fire two separate attacks. We can assume they are separate attacks because the wording references taking a penalty on "each shot".
If both barrels are fired at once, they must both target the same creature or object, and the gun becomes wildly inaccurate, taking a –4 penalty on each shot.
Vital Strike also modifies the attack action, making it explicitly clash with the double barrel rules. It refers to making "one attack at your highest BAB", which is clearly incompatible if the above rule regarding double barreled weapons is true. As a result, I would rule that you can use either modifier to the attack action separately, but not together.
Again, your DM's interpretation may vary.
From the musket description:
each barrel can be shot independently as separate attacks, or both can be fired at once as a standard action (the attack action).
Vital Strike can definitely be used for both one shot and two because the musket description says it uses the attack action in both cases.
From Vital Strike:
add the results together before adding bonuses from Strength, weapon abilities (such as flaming), precision-based damage, and other damage bonuses.
Vital Strike is not precision damage, it is applied before precision damage as stated in the feat description. Because it is not precision damage, Vital Strike applies to both bullets.