Does the extra 3d6 piercing damage of an Oathbow get doubled if the attack on the declared enemy is a critical hit?
Yes, according to the PHB (p. 196) and basic rules, critical hits double the damage of all dice tied to an attack's damage (emphasis mine):
When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together. Then add any relevant modifiers as normal. To speed up play, you can roll all the damage dice at once.
For example, if you score a critical hit with a dagger, roll 2d4 for the damage, rather than 1d4, and then add your relevant ability modifier. If the attack involves other damage dice, such as from the rogue's Sneak Attack feature, you roll those dice twice as well.