Stylistic level of a phrase starting with an infinitive[?] form

Solution 1:

There are several issues.

Firstly, some simple ones: it should be publication in a journal, not on; and the process as a whole is usually called peer review, not peer reviewing. [Edit: a third "correction" removed after Colin Fine convinced me it's not needed.]

Secondly, the form …requires to go through… is incorrect. Requires is never followed by just an infinitive in English — it can be followed by either a gerund (the necessary action), or a noun phrase plus an infinitive (the person/thing who has to do the action, plus the action):

Joining the army requires passing a physical exam.

Joining the army requires you to pass a physical exam.

Thirdly, as Adam points out, there’s a slight clash in your sentence between the verbs bringing (whose subject is the author) and go through (whose subject is the paper itself). Given the form of the sentence — the verbs aren’t literally parallel — this isn’t quite a matter of grammar (to my ear) so much as of meaning. Something like

Bringing an article from draft to polished publication in a scientific journal requires it to go through a process called “peer review”.

is grammatically quite correct, parallel to eg:

Sending your daughter into the army requires her to pass a physical exam.

But it’s a slightly awkward phrasing, making a shift in agent that really isn’t necessary. So in the end I’d suggest something like:

Bringing an article from draft to polished publication in a scientific journal requires putting it through a process called “peer review”.

Solution 2:

As is, I don't think that your sentence is acceptable. If you are going to start with an active word like 'bringing', you should have the agent that is doing the bringing somewhere:

Bringing an article from draft to polished publication in a scientific journal requires that the author subject it to a process called "peer reviewing."

I think that I would leave off the word bringing and choose a wording more like this:

For an article to go from draft to polished publication in a scientific journal, it must go through a process called "peer reviewing."

Or even better:

For an article to go from draft to polished publication in a scientific journal, it must go through "peer review."

To answer your other questions:

• It is not really possible to say what 'to go' is here, but it is not used correctly in any sense. Your sentence's subject is the gerund 'bringing,' and the verb is 'requires,' but 'to go through...' is not a workable direct object.
• I believe that it is not stylistically or grammatically acceptable