In general, one may not ask for one's personal needs on Shabbath. There are various reasons for this, one of them being that one may become sad and cry on Shabbath.
Included in this is praying for one who is ill. Even though it is commonly done nowadays, nevertheless, there is a difference of opinion among the Posqim and there are those who rule that one may not recite a Misheberakh (blessing) for a Holeh SheEino Mesukkan (one who’s life is not in danger) on Shabbath. If reciting a Misheberakh for one who’s life is not in danger, it is appropriate to minimize the blessing to "May G-d have mercy upon him among all the sick of His people Israel". It is appropriate to add the sentence: "Shabath Hi Miliz'oq etc. [hear in full on the audio]"
In the case of a Holeh Sheyesh Bo Sakkanah (one who is critically ill - whose life is in danger), however, all agree that a blessing may be recited, on Shabbath, as is mentioned in the Shulhan 'Arukh. A woman who is having difficulties during labor, is considered to be in this category.
(See Shabbath 12 a&b. Sh. 'A. 287: 1 & 288: 10. Kaf Hayim 288: 76, 78, 80. M.B. ibid, 28.)
