Vivian is an inanimate soap bar that Snaptrap carved when he was in prison. Even though Vivian was never really alive, Snaptrap (and some others) talked about her as if she was alive.
She "spoke" in a Southern accent (ventriloquized by Snaptrap) and said "y'all" on occasion. Snaptrap would always be holding her in one hand and would move her from side to side. As revealed by Snaptrap, Vivian's "dream" was to be soap for a hotel's men's room.
Vivian was only seen in "Mall Rat," because when Snaptrap set the mall heading for the Sun, she melted under the heat.