Why does the Holy Qur’an call Adam (as) and David (as) in particular a vicegerent of God?
Every prophet is a vicegerent of God. Adam (as) and David (as) are called this because neither of them took over prophethood from an immediate predecessor. God called each of them directly.
In essence every prophet is a vicegerent of God on earth. He represents God’s will among people and guides them on His path. Even so, the Holy Qur’an uses this title explicitly for only two prophets, Adam (as) and David (as). There is a reason for this.
Adam (as) inherited his prophethood from no one. Before him there was no prophet whose successor he could have become. God called him directly, and that is why he is called a vicegerent in so many words.
With David (as) it is similar. He appeared among the Israelites after there had been no prophet among them for some time. So he did not simply take over the legacy of a predecessor. God Himself prepared him for his task and appointed him directly. That is why the second verse above addresses him by this title.
One might ask why, then, the Holy Prophet Muhammad (saw) is not described in the same way. Among the Arabs, too, there had been no prophet for a long time before him. The answer goes deeper. When God spoke to the angels and announced a vicegerent on earth, it was above all he who was meant. With Adam (as) God’s vicegerency began in a small way, and in the Holy Prophet Muhammad (saw) it reached its fulfilment.
So the first verse above is not only an account of the beginning of humankind. It points to the summit towards which this story was one day to lead.