A vanity is a dressing table with a mirror. Vanities are located in the furniture section, all the way to the left in Design Mode. Then, choose the subcategory all the way to the right. The choices for vanities are as follows: Matilda, Francesca, Sherry, Maison, Petal, Jerri.
Click the link below to see an image showing where the vanity section is.
Hope this helps!