This round plastic object is inserted horizontally in the wood socket, then rotated and allows tightening the screw to fix together two pieces of my bed.
What is this hardware called? Can a replacement be purchased independently?
The hardware you are looking for is called a "cam lock." The round, locking piece is the cam lock nut. It uses a corresponding bolt called, naturally enough, the cam lock bolt.
available at most major big box home centers in the hardware dept. They are usually found in the rack of trays used for the myriad of misc hardware bits and pieces