According to the Gemara in Brachot, there are two views about Shacharit, Mincha, and Maariv: one is that the Avot instituted them, and one is that they correspond to the daily tmidin and the burning of parts on the mizbeach at night.
Who instituted Musaf? Clearly it corresponds to the korban musaf, but when did people start davening it?
(One can similarly ask about when did people start davening Shacharit, Mincha, and Maariv, according to the view that they correspond to the korbanot.)
The Sefer Birchei Yosef 423:2, which is written by the Chida says that Musaf for Rosh Chodesh was instituted by Rochel Imainu. Her name is hinted in the first letters of Ro'shei Cho'doshim L'Amecha.
The Ramat Shmuel in his commentary on Pesachim writes that Musaf was established by Avraham.
Since the posuk (Bereishis 19:27) says: "And Abraham got up early in the morning to the place where he had stood before the LORD." which implies that he had stood there previously to pray and if so, necessarily we are talking about Musaf. This is why it said "to the place where he had stood there" to pray Musaf after he prayed Shachris, since Musaf comes right after Shachris.
Clearly? It is not at all clear that it corresponds to the korban musaf.
As Elbogen writes: "But the earliest sources know of an Additional Service even without a sacrifice. It is reported of the ma'amadot that they would hold services four times on each weekday; one of these services was called Additional (compare M. Taanit 4:1,4). Perhaps it is to this service that the dictum of R. Eleazar b. Azariah refers: "The Additional Service takes place only in a town assembly" (M. Berachos 4:7), according to which the institution of this service is conditional upon the existence of a communal association -- that is, one that participated in the ma'amad. It would follow that originally, the Additional Service had nothing to do with the Additional Sacrifice. But all the other sources in our possession know of an Additional Service only on days when the Additional Sacrifice was brought (compare Num. 28,29). It is first mentioned by first century tannaim (T. Berachos 3:3; compare 3:10-11; T. Sukkah 4:5)
