All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Deliberate vomiting invalidates the fast. It is impermissible for a fasting person to deliberately make himself vomit unless he is sick, in which case he may vomit and make up for that day later on. As obvious in your question, your sister was not sick in such a way that made her need to vomit. Rather, she used to do so only for the sake of feeling relief. That being said, your sister is required to make up the fasts that she broke by her deliberate vomiting and ask Allah to forgive her. In addition, she has to pay an expiation for each day of fasting that she delayed until the following Ramadan came.
Allaah Knows best.