You'll start this Chapter by getting a new party member -- Shade! She's a mandatory party member for this next section, so choose the rest of your party, then head left to find a Chao Egg. Go back down the hill on the left and Fly southwest. Jump to the round platform, then Jump again into the hole. Interact with the first generator, then leave.
Head over to the structure on the eastern side of the zone and scale it to the roof via the springboards. Use the ramp there to reach the crumbling building northwest of it, and take the Chao Egg there. There's another Chao Egg on top of the westernmost structure in the zone as well, so go retrieve that.
Now, for those generators. The second one can be found through the doorway in the southeast corner, east of Eggman 's Lab. Go activate it. The third generator can be found up the ramp from the second generator; be sure to take the Chao Egg in there as well.
Now that Eggman's lab's electricity is flowing once more, head to the upper room of the eastern building. Save your game, then go through the door east of the ramp. This is the Hazard Vault, and you'll be fighting a boss here!
The Egg Bot has two Assault Drone MK. IIs with it, and any round where the Drones are still alive, the Egg Bot can repair itself for 30 HP. Though it may sound like you should go after the Drones first because if this, you should not neglect inflicting damage on the Egg Bot, but its attacks can be devastating. In particular, his Shocker attack (no, not that Shocker) can hit you with serious damage as well as stun you, so you'll want to make sure you defend properly against that by hitting the prompts. Try to use attacks that hit all three enemies at the same time to spread the damage around, and use your very strongest POW Moves on the Egg Bot.
For defeating the Egg Bot, you'll receive the Dimensional Buffer. Head right and Smash through the crates there to find a Chao Egg and a box containing a Cure All Spray, POW Drink, and Iron Claws. Exit the Hazard Vault now, Jump down twice, then return to the lab so you can talk to Tails and end the Chapter.