Speculation
The Necromancer / Sauron
If we presume "Morgul" as the term used to be linked with Minas MorgulMinas Morgul, which thus implies a domain / strongholdstronghold of Sauron, then we can look to early drafts of The Fellowship of the RingThe Fellowship of the Ring:
Gandalf calls the weapon that was used 'a deadly blade, the knife of the the Necromancer which remains in the wound', not a 'Morgul-knife'
The Return of the Shadow: The History of the Lord of the Rings, Pt1, Christopher Tolkien - p.211
-p.211
In a later draft:
The 'Morgul-knife' is still the 'knife of the Necromancer'
The Return of the Shadow: The History of the Lord of the Rings, Pt1, Christopher Tolkien - p.363
p.363
Thus - we can see that originally, Tolkien was associating the one named weapon in the text very directly with Sauron. He later changed it of course to 'Morgul-knife' but again, given the association of Minas Morgul and Sauron, it isn't unreasonable to presume then that Sauron or someone quite close to him constructed their weapons.
Source: The Return of the Shadow: The History of the Lord of the Rings, Pt1 - Christopher Tolkien