The Gospels make no mention of Elizabeth as Mary’s midwife (nor do they mention any other midwife, either). There is a local midwife who appears in the Protoevangelium of James, the apocrphyal text written about 150 AD; this text, though, doesn’t have the authority of Scripture, so it’s hard to look upon it as a completely reliable source, especially when its account of the birth of Jesus diverges in several ways from what we find in the Gospel infancy narratives.
It seems fairly unlikely that Elizabeth would have been Mary’s midwife, given that she is not mentioned at all in the Gospel accounts of Jesus’ birth. The Mary and Elizabeth relationship is given so much emphasis in the Gospel of Luke (see the story of the Visitation in Luke 1:39-56) that if Elizabeth had been present to help her cousin deliver, it seems logical to assume that Luke would have mentioned it. In the end, that’s probably about as close to an answer as we can get.