James 5:14-15 Is any one of you sick? He should call the elders of the church to pray over him and anoint him with oil in the name of the Lord. And the prayer offered in faith will make the sick person well; the Lord will raise him up.
While I would be the first person to raise my hand and state a normal, healthy, laboring woman is not sick, I do see a use for anointing with oil at some births.
Not all labors happen without incidence. Sometimes blood pressures rise or heart rates seem to drop. Sometimes things seem to go haywire for no good reason. Sometimes the mother is overwhelmed with fear. Usually these are not emergency situations that require immediate action, and the family can be allowed ten to fifteen minutes to consider their options.
It is in these times that I would suggest anointing the woman with oil and praying over her. If there is a real problem, God can fix it faster than anyone else anyway. If it was not a real problem, but a case of impatience by caregivers, God can fix that too.
There is no negative effect in praying over a woman in labor, except that hospital staff may start referring to you as “that woman.” It can provide a laboring mother with comfort and encouragement. And once again, while I do not suggest a need for anointing during a normal birth, I do believe it is a valid option if a problem develops or if the mother finds it comforting.
Keep some oil in your labor bag anyway. If you need it for anointing, you know it is there. If all goes normally, you can use it for back and neck rubs.