Sort of. In the years leading up to and in the years following Charlemagne’s coronation in 800, the ‘vacancy’ of the imperial throne in Constantinople was used as justification for the decision to revive the imperial title in the West.

Leading up to the coronation, we don’t have much in the way of documentary evidence that suggests the Carolingian court or Rome were actively thinking about an imperial coronation. One of the few exceptions is in a letter Alcuin sent to Charlemagne in the summer of 799:

Up to this time there have been three persons of highest standing in the world. The first was that Apostolic Eminence who rules Vicar the See of blessed Peter—and what has been done to its holder in these times, your courtesy has told me. The second is the imperial dignity and secular power of the Second Rome [Constantinople]— and its governor has been cast down, by its own fellow-citizens, as is being told everywhere. The third is that royal dignity in which the will of our Lord Jesus Christ has made you rule of the Christian people—you surpassing in power these aforesaid dignities, outshining them in wisdom, standing even higher than they in the glory of your Kingship. See, all hope of the churches of Christ rests in you alone. To you it falls to avenge crimes, to guide the wandering, to comfort the distressed, to lift up the good. (emphasis mine)

The conventional reading of the letter is that Alcuin is making the case for Charlemagne’s claim to the imperial title. It’s worth pointing out that the conception of imperial authority had transformed over late antiquity and the early medieval period, and was increasingly bound up with the claim of divinely appointed authority over the Christian community. The coronation in that sense can be read as a rejection of Eastern Roman imperial and religious preeminence over the West and the Holy See (Rome had long been within Constantinople’s sphere of influence). Alcuin is here making the case that with the Eastern emperor out of commission and the pope recently expelled from Rome, it is up to Charlemagne to take control of the situation and fulfill the role of the good Christian emperor. Irene is conveniently ignored.

In the Annals of Lorsch, for which we have fragments of entries for 799 and 800 dated soon after the events they describe, the argument is more explicit.

Since there was no longer an emperor in the land of the Greeks and they all were under the domination of a woman, it seemed to Pope Leo and to all the fathers who sat in the assembly, as well as to the whole Christian people, that they should give the name of emperor to the king of the Franks, to Charles, who occupied Rome, where the Caesars had customarily resided, and also Italy, Gaul, and Germany. Because Almighty God had consented to place these lands under his authority, it seemed right, according to the desire of the whole Christian people, that Charles should also bear the imperial title. (emphasis mine)

The argument that emerges from early accounts of the coronation is that the 'vacant' imperial throne in the East, coupled with Charlemagne’s control of traditional imperial lands in the West, were sufficient justification for Charlemagne’s claim to the title. This argument differs from later accounts of the coronation which give greater significance to the pope’s role in the process as a source of justification. It is one of the defining differences between the Annals of Lorsch and Einhard’s later account in the Life of Charlemagne that in the former the assembly of clergy and laity plays a greater role in the acclamation of Charlemagne as emperor.

Which brings us to Pope Leo’s decision. The above arguments emerge from the Carolingian court in the run up to and years after the coronation. We do not have similar documentation that reveals the pope’s thinking on the matter, and as indicated above, there may be reason to think that the pope played less of a role in the coronation than later accounts would have us believe. However, given the pope’s tenuous hold on the See in 800, it is possible that Charlemagne’s supporters would be hesitant to hitch their wagon to an unsteady star, and therefore make their case for the throne on other grounds.

The ‘vacant’ throne argument is less a motivation and more an opportunistic argument that bolstered Charlemagne’s claim to the imperial title. The extent to which the argument was needed to convince Leo III or the leading citizens of Rome or expedited their decision is unknown. For their part, the Romans in Rome seemed more than willing to confer the imperial title on Charlemagne.