Would I be able to say, "I work for the newspaper" ?
If not, would I be able to say that using 'pentru' or is a different word required?
There is a slight difference in meaning.
"Eu lucrez pentru ziar" would imply that I'm an employee of the company owning the newspaper.
"Eu lucrez la ziar" would mean that the newspaper is a project I'm working on, either being paid for it or not. For example, say I'm a drawer; I might draw something for an article that is to be published in a newspaper or I might draw a book cover; then I can say which task gets my attention at the moment, using "la".
Romanian has a rule where you don't use the "the" form of a noun after a preposition (which is what "la" is). The exceptions are the preposition "cu" and when the noun is modified by an adjective.
Think of it like an extension of things like "at home", "at work" in English.