The Ever Valorous Regalia - Hammerhead
First Available: Chapter 3
Other Requirements: Complete The Ever Gleaming Regalia
Rewards: 1,000 EXP & Supercharger Component
This is another one of the quests in the long chain from Cindy and despite being able to accept it as early as Chapter 3, you'll be unable to complete it until Chapter 7 or later. It takes place in the Vesperpool area which is inaccessible until the story takes you there midway through the game.
Once you have access to this area, you'll find the item you're looking for nearby a dirt road next to an old abandoned car and shack. If you still need help finding it then use my screenshot below for further guidance. Your reward for this quest will be the Supercharger Component which will increase the Regalia's top speed.