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Is the Sneak Attack damage dealt by a splash weapon multiplied by Energy Vulnerability?
Underground Chemist is a Rogue archetype which swaps some Rogue abilities with Alchemist ones. Their "Precise Splash Weapons" ability says:
At 4th level, an underground chemist can deal sneak attack damage with splash weapons.
If I use a splash weapon with an energy type (fire, acid, etc.) against a foe with energy vulnerability, does my sneak attack damage get multiplied by 50% also?
Precision damage is the same type of damage as the weapon that applied it. If you sneak attack somebody with a bludgeoning weapon and they have resistance, then the grand total of your damage is reduced. The same should apply in your case with vulnerability. I think there are some small cases (such as ability damage) where this doesn't apply.
James Jacobs, Pathfinder's Creative Director, has this to say about precision damage (The link):
Seems to me that the notion of "precision damage" might be in the same category as "iterative attack,"—a useful game term that for whatever reason is never defined. And since it's not defined, you see the word precision pop up as descriptive text. Precision damage is, basically, extra damage caused by placing a blow in a precise spot where the damage is more damaging than a simple blow. That's basically the same thing that's going on with a sneak attack or a duelist's precise strike, and it's also what's going on with a critical hit.
That's probably as close to a ruling as you're going to get at the time. You can most likely infer that itself the same type of damage, but it's not explicitly stated either way. If you're looking for RAW, I'm not sure it exists at this time.
