Upon entering Longneck Lukowski's Cannery, you'll see Trader Rylee and Theodore arguing. You can talk to Rylee before she leaves and ask her about the sick people and her vault suit. (If you haven't visited Vault 81 yet, she'll mark it on your map) From there, you can also invite her as a new settler to one of your settlements.
Supposedly, you can ask Theodore immediately after Trader Rylee leaves but sometimes it will cause a dialogue bug where that option is missing. To fix this, head to the SE corner of the cannery then take the lift down. This will trigger the quest. The lift is locked from going up so use the terminal to open the next door.
Before taking the lift down, make your way to the elevated control room in the upper level of the cannery. It has a Novice terminal that can open the security door beside it. (Unfortunately, this gate only contains a Short Combat Shotgun. In the same level, there's a Weapons Workbench.
Near the weapons workbench, you can find a red steamer trunk, Barter Bobblehead, and an issue of Tales of a Junktown Jerky Vendor.
The door has radroaches, chem station, locked novice ammo box, and a tesla trap while the corridor further north has feral ghouls. Kill the ghouls then go through the Hole in the Wall to further explore the basement area. Expect more ferals in the first hall then go through a tunnel to reach the second hall.
There are more ghouls hiding in the Water so be ready. It will help targeting the area using your VATS so you can kill them pre-emptively. Continue to the next room then go down the flooded floor. After killing the ghouls, there's a flooded tunnel to the west behind the stairs that leads to room with a duffel bag, advanced safe, and suitcase.
Backtrack then follow the tunnel now with more ferals. The end of the tunnel to the north is a dead end so take the first turn to the right, you'll end up in another large but irradiated hall with more ferals, including a glowing one. Loot the duffel bag here then follow the tunnel south.
You'll reach another large hall. Scan the area for hiding ghouls, one on each corner. Cross the room and take the stairs to reach the walkways above. The path to the north leads to a lone turret with a novice ammo box. Continue east and you'll reach a generator with fusion core.
The next room and the following paths are filled with traps so be careful. Past the trap-laden walkway is a room with duffel bag, novice first aid box, and a turret on the stairs. Finally, continue upstairs to find a small room with mutilated ghouls and a chained door. Remove the chain and to get back to the main cannery building. Theo will become hostile as well as every cannery robot. You'll be forced to kill him to complete the quest.