Donald J. Trump for President, Inc., along with the Republican National Committee and the New Jersey Republican Party, filed suit Tuesday.
The campaign says Gov. Phil Murphy's vote-by-mail order violates citizens' right to vote.
It also says the governor misappropriated power that belongs to the state legislature.
Murphy says keeping most polling places closed will help stop the spread of the coronavirus.
His order follows the model the state used in its July primary.
Murphy said Friday all registered voters would get a ballot.
Murphy said voter can mail their ballots, use an official drop box, or drop them off in person at county offices or a polling place.
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Voters arriving to vote in-person will be casting provisional ballots.
Ballot drop boxes have been used with success for several years in states like Oregon, Washington and Colorado that rely largely or entirely on ballots that must be sent in.
But their use is being expanded because of the coronavirus outbreak and, more recently, concerns about the post office's ability to do its job.