We only know of a single Lieutenant of Morgul, Gothmog, who held the rank during the War of the Ring. We can therefore only be absolutely sure that a Lieutenant existed at the end of the Third Age, but it seems reasonable to assume that there were other Lieutenants dating back into history. The earliest possible date for a Lieutenant of Morgul to have existed was III 2002, the year in which the Nazgûl captured Minas Ithil and made it their own city of Minas Morgul.

We know that Gothmog held the position of Lieutenant during the War of the Ring, but we're never told what kind of being Gothmog was. Potentially he might have been one of the Nazgûl and, as an undying servant of the Ring, might therefore have held the rank since Minas Morgul was captured. There are, however, reasons to think this unlikely. More plausibly Gothmog might have been a Man (especially a Black Númenórean like the Mouth of Sauron) or an Orc. Of course, if the rank of Lieutenant was held by a mortal being like this, there is no reason the race of its holder would need to be fixed: over the history of Minas Morgul, an Orc Lieutenant might at times have been succeeded by a Man, or vice versa.