Prey’s "The Corpse Vanishes" optional objective is one of the first side missions you’ll pick up. You’ll get it shortly after starting the game — before you even know the truth about your world — but you won’t have access to the area where you can complete it until almost halfway through the game.
Jovan Gravilovic’s email
You’ll pick up the "The Corpse Vanishes" optional objective from an email you read shortly after escaping from the simulation lab. During your escape, you’ll pass through a decontamination chamber (and watch mimics kill someone). Jovan Gravilovic’s cubicle is the first one on the left in the offices you enter right afterward. He’s got an email with the subject "URGENT!"
The email will set you along to path for this objective. It directs Jovan (and now you) to go get Bellamy’s cabin keycard off of his body which is in the trauma center.
Search for Bellamy’s corpse in the trauma center
The trauma center is one of the optional rooms to explore in the Talos 1 lobby. There’s a keycard for the trauma center at the desk behind the elevators in the lobby, or there’s one on Bianca Goodwin’s desk up in the executive offices on the third floor (two doors down from your office).
There’s a lot going on in the trauma center, but for the purposes of this objective, you only need to go as far as the first desk. You’ll find two things here for this objective (only one of which is helpful): Dr. Sylvain Bellamy’s tracking bracelet, and an email saying that he was taken to psychotronics instead of the trauma center (specifically, the morgue).
Get Bellamy’s keycard from his body
You’ll get access to the morgue during the main story objective "Detour." Shortly before you get to the GUTS, you’ll find yourself in the live exam room. Take the grav lift down to the morgue, where the bodies are is difficult to get to. You have to smash a window and use your huntress bowcaster to open the door.
Bellamy’s corpse is on the gurney ahead of you when you walk in, and you can loot the Sylvain Bellamy’s cabin keycard from his corpse.
Use Bellamy’s to access his cabin in crew quarters
After completing "Detour," you’ll end up in the crew quarters area of the arboretum. Bellamy’s cabin is very close to the main lobby. When you enter, head past the concierge desk, then turn left and go down to the crew cabins. Take the hall on the right, and Bellamy’s cabin is the first door on the right.
When you’re inside, download the file off the computer to complete "The Corpse Vanishes." Doing so unlocks a new typhon neuromod power: lift field.