Why is the virgin goddess Artemis associated with childbirth?
Orphic Hymn I to Prothyrea ("Προθυραία"), identifies Artemis as the goddess of childbirth, and relegates Eileithyia to the role of "assisting goddess":
O venerable goddess, hear my pray'r,
For labour pains are thy peculiar care;
in thee, when stretch'd upon the bed of grief,
The sex as in a mirror view relief.
Guard of the race, endued with gentle mind,
To helpless youth, benevolent and kind;
Benignant nourisher; great Nature's key
Belongs to no divinity but thee.
Thou dwell'st with all immanifest to sight,
And solemn festivals are thy delight.
Thine is the talk to loose the virgin's zone,
And thou in ev'ry work art seen and known.
With births you sympathize, tho' pleas'd to see
The numerous offspring of fertility;
When rack'd with nature's pangs and sore distress'd,
The sex invoke thee, as the soul's sure rest;
For thou alone can'st give relief to pain,
Which art attempts to ease, but tries in vain;
Assisting goddess, venerable pow'r,
Who bring'st relief in labour's dreadful hour;
Hear, blessed Dian, and accept my pray'r,
And make the infant race thy constant care.
The Hymns of Orpheus, translated by Thomas Taylor
Why would the Greeks associate a virgin goddess with childbirth? Is the association limited to the Orphic tradition?
It is commonly accepted that she was born first and assisted with the birth of her brother Apollo.
Two quick sources I pulled off of Theoi:
"Of the daughters of Koios . . . Leto had relations with Zeus, for which she was hounded by Hera all over the earth. She finally reached Delos and gave birth to Artemis, who thereupon helped her deliver Apollon. Artemis became a practised huntress and remained a virgin." Pseudo-Apollodorus, Bibliotheca 1. 21 (trans. Aldrich) (Greek mythographer C2nd A.D.)
There is another explanation regarding the painless labor of Leto:
"Even in the hour when I [Artemis] was born the Moirai (Fates) ordained that I should be their helper [women in childbirth], forasmuch as my mother suffered no pain either when she gave me birth or when she carried me win her womb, but without travail put me from her body." Callimachus, Hymn 3 to Artemis 22 ff (trans. Mair) (Greek poet C3rd B.C.)
However, these two sources may be later attempts to reconcile the association of the virgin form of Artemis with childbirth. In earlier forms, most famously at the Temple at Artemis at Ephesus, she is depicted in a manner strongly associated with fertility, and may have been a mother goddess until her status changed to that of virgin daughter in the Olympian pantheon.
In terms of limitation to the Orphic tradition, I don't know if there is a definite answer in that this association was almost certainly more widely held as a folk belief.
I have a personal theory, based on Clytemnestra's dialogue in Iphigenia at Aulis, that the insult to Artemis by Agamemnon actually arose out of his status of "baby killer" in that he killed Clytemnestra's infant after murdering her former husband to take her by force in marriage. Euripides also probably uses this to reinforce the idea that her subsequent slaying of her husband is rooted in these acts, not jealousy over Cassandra, per Aeschylus.