Susannah is Mrs. Shandy's maid and a walking Calamity Jane. We don't get a ton about Susannah's personality, but we do learn three things about her function in the story: she supports Mrs. Shandy's desire for a midwife by running to get midwife instead of the doctor, as Obadiah sees in 1.6; she garbles Trismegistus into Tristram; and she aids and abets Tristram's circumcision.
All these acts have one thing in common: she slyly undermines Mr. Shandy like it ain't no thang, all the time. Better yet, she stays cool in the face of patriarchal authority. She even orders the doctor around, flat out telling him that he doesn't need to bother looking at the midwife's bruised hip but that he "had better look at [her] mistress" (3.13.1).
She's also one smart cookie—no one gets by her, even an anxious husband. Mr. Shandy never goes directly in to see his wife giving birth; instead, he has to call Susannah when he sees her "passing by the foot of the stairs" (3.22.1) and then asks her again, "how does your mistress?" (4.14.9) after the baptism. She does the honor of carrying Tristram's name to Yorick, running speedily along the corridor and forgetting the name as she goes (4.14.3).
Even when Susannah's not trying, she messes with Mr. Shandy's head. She changes Tristram's name and is responsible for his circumcision, thus bringing Mr. Shandy's nightmares to life. And you can bet Mr. Shandy blames Susannah for the chaos: he twice refers to the circumcision as "Susannah's accident" (6.13.2, 6.14.2). Since Tristram ends up being the heir to the Shandy family, the accidents basically undermine the whole patriarchy. You go, girl!