All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If the woman in question was overcome by vomit and she swallowed nothing of the vomit that she was able to spit out, then this should not invalidate her fasting. Therefore, she is not required to make up the fasting. If she intentionally vomited or was forced to vomit and swallowed some of the vomit back, even though she held it in her mouth and was able to spit it out, then her fasting is invalid and she has to make up for the days of fasting in which she went through this.
The woman has to make up for those days before the following Ramadan comes. If she is unable to make up for those days before the following Ramadan comes because she fears for herself or for her child and her fears are significantly justifiable under the Sharee’ah, then she may delay fasting those days until she is able to. In this case, she does not have to offer an expiation because she has a valid excuse for the delay. She is required to start fasting whenever her excuse ends.
Allaah knows best.