A good source for election-related vocabulary will probably be the Commentariolum Petitionis, written by Quintus Tullius Cicero for the benefit of his more famous brother Marcus ahead of the latter's successful bid for the consulate (presumably, but the authenticity is questionable—it is possible the text was actually written by an unknown post-Augustean author).
Going by the title, a “bid for office” or “candidacy” can be called a petitio.
The office you run for goes in the genetive: petitio magistratus, tribunatus, aedilitatis and so on. One who runs for an office (qui petitioni se dat) might be called a petitor (supposedly the more usual word would be candidatus, although the above-mentioned Commentariolum uses petitor a few times). His competitors are called, well, competitores.
If we think of an “election campaign” as the very process of attempting to be elected to an office, then I think this is a good word, but in English we might prefer translations like “bid, application, candidacy.” If we think more of “canvass,” then I believe ambitio, as cited by Joonas, would be more appropriate.