Does she have to do wudoo’ because of vaginal discharge in order to read Qur’aan?

Firstly: 

It is permissible for a
person who is in a state of minor impurity to read Qur’aan without touching
the Mus-haf, and it is permissible for him to touch books of Tafseer or to
read Qur’aan from them. But he is not allowed to touch the Mus-haf unless he
has wudoo’. See the answer to question no.
10672. 

Secondly: 

Vaginal discharge is taahir
(pure), but there is a difference of opinion as to whether it invalidates
wudoo’. The majority of scholars are of the view that it does invalidate
wudoo’, but if it is continuous, then it comes under the same ruling as
urinary incontinence, so the woman should do wudoo’ for every prayer and
then offer whatever obligatory and naafil prayers she wants, and touch the
Mus-haf, and it will not matter if this discharge comes out. 

But if the discharge is not
continuous, then you can read from the Mus-haf by putting a barrier over
your hand between you and the Mus-haf, such as gloves and the like. 

See the answer to question
no. 2564. 

And Allaah knows best.