I think 'she writes for newspapers' ought to be accepted, as in English if we're talking about newspapers in general, we don't put the article. I get confused as to when Spanish sometimes puts in articles and sometimes doesn't. It is hard to tell in these isolated sentences whether we are talking about a generalisation or whether we are talking about specific newspapers.
In Spanish, unlike in English, a common-noun subject requires an article; for general statements about a subject, the subject takes a definite article. So, for example, ‘Los periódicos consumen mucho papel.’ can mean either “Newspapers consume a lot of paper.” or “The newspapers consume a lot of paper.”. A common-noun object, on the other hand, does not require an article, as in ‘Reciclamos periódicos.’ = “We recycle newspapers.”.
In both Spanish and English, the newspaper publishing industry is indicated by the definite plural ‘los periódicos’ = “the newspapers”, just like the banking industry is termed ‘los bancos’ = “the banks”.