What is this headless screw called?

In my neck of the woods it would be a "set screw". You may not find one with the exact cross pattern in the top (most are Allen or slotted), but it should still work. The important parameters are diameter and thread pitch.

I don't see those values specified in that document, so some trial-and-error may be required. If possible, take the part it engages to a hardware store and test with common screws for size, then look for a set screw with the same diameter and pitch. A properly-fitted screw will turn easily until it snugs against the surface below.

Call the company and tell them you're missing that part number.

Anything which jeopardizes the safety of a baby is usually handled promptly and without question as companies generally wish to avoid onus in the event of an accident.