Does this refer to the laying of sins on the scapegoat (i.e., covering it with the sins of the people) and, if so, why should this be termed “to make an atonement with him”?
Yes, I believe this referred to the laying of sins on the scapegoat. Leviticus 16:20-22 explains that Aaron put on the head of the goat all the iniquities, transgressions and sins of the people of Israel and sent the goat free into the wilderness.
Lev 16:20-22 “And when he has made an end of atoning for the Holy Place and the tent of meeting and the altar, he shall present the live goat. (21) And Aaron shall lay both his hands on the head of the live goat, and confess over it all the iniquities of the people of Israel, and all their transgressions, all their sins. And he shall put them on the head of the goat and send it away into the wilderness by the hand of a man who is in readiness. (22) The goat shall bear all their iniquities on itself to a remote area, and he shall let the goat go free in the wilderness.
This could be termed “to make an atonement with him” because the scapegoat was used to symbolise the removal of the sins of the people of Israel. It pointed forward to the atoning work (Psalm 103:12) God planned to bring about through his son (John 1:29), when Jesus would lay down his live for us and cleanse us from all our sin (1 John 1:7).
There is a very interesting little book called “Law and Grace – A devotional study of the Law of Moses”, by W. F. Barling. I’ll quote a section from this book from pages 146-7, which is relevant to the question.
… the ritual then continued with particular reference to the live goat, this being symbolically its dead partner acting in a different capacity, since the same animal could not both forfeit its life to show the means of expiation and also serve to illustrate its effect. It fell to the live goat to illustrate the effect of expiation in the place of its dead partner. “Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel and all their transgressions in all their sins, putting them upon the head of the goat, and shall bear upon him all their iniquities unto a land not inhabited” (Lev. 16:21-22).
The goat – by now the embodiment of Israel’s sins in the year just ended – was then taken away from the Sanctuary and Camp that is, from the presence and face of God. The ritual was transparent. The expiation effected by the blood of the first goat was shown by the departure of the second to have effected the utter removal of all Israel’s sin from before God (Lev. 16:22), the scope of Lev. 16:21 being far too comprehensive, as we have seen, for us to limit the expiation just to sins so far unatoned for.