The fishbowl helmet is obtained from Murphy in exchange for a fishbowl during the Recipe for Disaster#Freeing Pirate Pete subquest of the Recipe For Disaster quest. It is used, along with the diving apparatus, to dive underwater to the Mogre Camp. The fishbowl helmet can be obtained from Murphy for free after freeing Pirate Pete without the exchange of a fishbowl.
In The Great Brain Robbery, this helmet is used with the diving apparatus, to dive in the underwater tunnel on Harmony Island.
If you lose the Fishbowl helmet, Murphy will give you a new one if you choose the diving option when talking to him.