It isn't exactly the Delphic oracle: it isn't the Pythia. The figure is the 'Delphic Sibyl', which is a mostly-christianised version of a mythical figure who originated as an antecedent to the historical Pythia.
The pendentives along the long sides of the ceiling depict twelve prophetic figures: seven of the prophets in the Hebrew Bible, and five Sibyls. The Sibyls originally come from a work by the ancient Christian writer Lactantius, who quotes a catalogue of ten Sibyls. He was in turn reproducing a catalogue by the pagan scholar Varro: Varro, Antiquitates rerum divinarum fr. 56a ed. Cardauns = Lactantius, Institutiones divinae 1.6.7-12. Several Greek-speaking Christian sources also give catalogues of ten Sibyls, but Lactantius is the most likely basis for Michelangelo's ones.
The reason Varro's pagan catalogue was enthusiastically adopted by Christians was largely thanks to a collection of fourteen Jewish and Christian hexameter poems in Greek called the Sibylline oracles, dating from the 2nd century BCE to the 4th century CE. These poems purport to be prophecies of various religious-related events, including things like the sufferings and destruction of non-Jewish nations, the punishment of the damned in the afterlife, and hymns and discourses about Christ.
Lactantius knew these poems (he quotes from them), and they were popular among ancient Jews and Christians: there are references to them in Josephus (late 1st century CE), the Shepherd of Hermas, and Justin Martyr; Clement of Alexandria claims that Paul consulted one of the Sibyls.
In addition, Vergil's Eclogue 4 (20s BCE) contains a prophecy of the birth of a heroic figure, made by the Cumaean Sibyl, which was at times interpreted as a prophecy of the coming of Jesus. This added to the Sibyls' street cred among Christians.
Michelangelo actually adheres to the sequence in Lactantius' list, though he only includes half the Sibyls: Lactantius' sequence is Persian Sibyl - Libyan Sibyl - Delphic - Erythraean - Cumaean, and if you start from the Persian Sibyl in the Sistine Chapel, make your way to the end, then jump back to the opposite end, they follow the same sequence. (But this may be a coincidence: it doesn't look like Michelangelo used any particular sequence for the Hebrew prophets.)
The Sibyls certainly originated in pagan Greek mythology, as fictive personas who supposedly composed 'chresmologic' poetry in the 6th-5th centuries BCE -- collections of supposed verse oracles. They get cited in this capacity by Herakleitos (fr. 92 Diels-Kranz), Plato (Phaedrus 244b), pseudo-Plato (Theages 124d), and for comic effect by Aristophanes (Knights 61; Peace 1095, 1115-1116).
But they have nothing to do with the historical Pythia at Delphi. It's thanks to the reframing of them in Vergil, Varro, and via the Sibylline oracles, that the mythical Sibyls were adapted and adopted by Christians.