The Whig Party had existed as a second party in the south prior to the rebellion. Though fewer in numbers, its adherents did provide an alternative voice.
Having one party during the rebellion may have helped provide consistent leadership, though a determination to protect the 'peculiar institution' and a new-found sense of nationalism overode any effort to form an opposition party.
Did that aid the rebel effort? possibly.
Did the existence of a two party system in the states that remained in the Union help or hinder the military effort? Not so much- at least early on. More troublesome was the anti-war faction, many of whom were Democrats. Fortunately, the Administration took pains to assure inclusion of pro-Union Democrats, so that opposition was minimized.
A greater problem within the Army was that so many high officers were politicians- of whatever party- and, except for some notable exceptions, ill suited to military command.