Results from a concerted Hubble Space Telescope (HST) survey of nearby white dwarfs for substellar objects are presented. A total of seven DAZ white dwarfs with distances of <50 pc had high-contrast and high spatial resolution NICMOS coronagraphic images taken to search for candidate substellar objects at separations 10'' away. Limits to unresolved companions are derived through analysis of Two Micron All Sky Survey photometry of the white dwarfs compared to expected fluxes based on the white dwarf's effective temperature, distance, and gravity. Our HST survey of seven DAZ white dwarfs identified candidate companions for four of the white dwarfs. For three of these, HST and ground-based second-epoch observations showed the candidates to be background stars. The fourth white dwarf, which is close to the Galactic plane, has seven candidate companions at distances of 2''–4'', which remain to be followed up. We find that for four of the white dwarfs we are sensitive to planetary companions of 10M_J. For all the targets, we are sensitive to companions of >18M_J. The lack of significant near-IR excesses for our targets limits any kind of unresolved companions present to be substellar. In light of these results we make several comments on the possibility of determining the origin of metals in the atmospheres of these white dwarfs.