She vomited involuntarily whilst she was fasting, and she thought that it was obligatory to make up that day, then whilst she was making it up she found out that her first fast has been valid; should she complete that day or break her fast?

If a person cannot help vomiting when he is fasting, then
his fast is still valid and is not invalidated by that. The evidence for
that from the Sunnah is the hadith narrated by at-Tirmidhi (720) from Abu
Hurayrah, according to which the Prophet (blessings and peace of Allah be
upon him) said: “If a person vomits involuntarily, then he does not have
to make up his fast, but whoever deliberately makes himself vomit has to
make up his fast.” Classed as saheeh by al-Albaani in Saheeh at-Tirmidhi.
We have discussed this in fatwa no. 38579. 

With regard to what you mention about having started to make
up these days, before finding out that your fast was valid and that you did
not have to make them up, the more correct scholarly view is that whoever
starts to do an act of worship, thinking that it is obligatory for him, then
he finds out that it is not obligatory for him, has the choice between
completing it or ending it, but completing it is preferable. This differs
from the view of Zafar, one of the Hanafis, who said that it is obligatory
for him to make it up if he stops it. It says in Badaa’i‘ as-Sanaa’i‘ fi
Tarteeb ash-Sharaa’i‘ (2/102), which is a Hanafi book: Our companions
differed concerning a fast which a person thinks he has to offer, if it is
broken, such as if he started a fast or prayer, thinking that it was
obligatory for him, then he found out that it was not obligatory for him, so
he broke the fast deliberately. 

Our three companions said: He does not have to make it up,
but it is preferable to continue with it. 

Zafar said: He has to make it up. End quote. 

In al-Jawharah an-Nayyirah ‘ala Mukhtasar al-Qadoori
(1/70), which is also a Hanafi book, it says: If he begins a prayer or fast,
thinking that it is obligatory for him, then he finds out that he does not
have to do it, so he ends it, he does not have to make it up in our view,
but Zafar said that he does have to make it up. 

If he begins to pray Zuhr, thinking that it is obligatory for
him, and a man starts to follow him in prayer, with the intention of
offering a voluntary prayer, then he remembers that he had already prayed
Zuhr, so he stopped praying, then he does not have to make it up and neither
does the one who followed him in that prayer. End quote. 

From this we know that you have the choice with regard to
completing that fast or breaking it, although completing it is preferable.

However it should be noted that this completion of the day’s
fast can only be with the intention of offering a supererogatory (naafil)
fast. 

It says in Kashf al-Asraar Sharh Usool al-Bazdawi
(2/312): If he starts a prayer or fast, thinking that it is obligatory for
him, then he finds out that it is not obligatory for him, then it becomes a
supererogatory act, according to scholarly consensus. 

If he ends it, he does not have to make it up later, because
of what we previously stated about him having the choice with regard to
whether he should continue. End quote. 

The view of Zafar, which is that he must complete it, is also
the view of the Maalikis. With regard to the matter of whether it is also
obligatory to make it up, if he ends it deliberately, they have two views,
but if he stops it by mistake, then he does not have to make it up,
according to consensus.

Please see Mawaahib al-Jaleel wa Haashiyatuhu (2/262)
and Minah al-Jaleel (2/153) 

with regard to liquid coming up from the stomach to the
oesophagus or mouth, this is what is known as acid reflux; we have discussed
in detail the ruling thereon with regard to fasting in fatwa no.
40696 

And Allah knows best.