Ben says that he does not vote a party line ticket. In the 1850s, "split ticket" voting, or voting for candidates of more than one party, was very difficult, if not impossible. Most voting was done with ballots which were pre-printed by the various parties and distributed to supporters. It would be another few decades before the secret or "Australian" ballot, which was provided by the government and listed all candidates, regardless of party, became popular in the United States.
While her father awaits election results, Lavinia and her friends discuss the Kansas-Nebraska Act and agree that, if they could vote, they'd support the Republican party. However, Edward Dickinson was elected to Congress in 1852. The Kansas Nebraska act was not passed till 1854, and the Republican party was formed afterwards.