Black holes with masses below approximately 10(15) g are expected to emit gamma-rays with energies above a few tens of MeV, which can be detected by the Fermi Large Area Telescope (LAT). Although black holes with these masses cannot be formed as a result of stellar evolution, they may have formed in the early universe and are therefore called primordial black holes (PBHs). Previous searches for PBHs have focused on either short-timescale bursts or the contribution of PBHs to the isotropic gamma-ray emission. We show that, in cases of individual PBHs, the Fermi-LAT is most sensitive to PBHs with temperatures above approximately 16 GeV and masses 6 x 10(11) g, which it can detect out to a distance of about 0.03 pc. These PBHs have a remaining lifetime of months to years at the start of the Fermi mission. They would appear as potentially moving point sources with gamma-ray emission that become spectrally harder and brighter with time until the PBH completely evaporates. In this paper, we develop a new algorithm to detect the proper motion of gamma-ray point sources, and apply it to 318 unassociated point sources at a high galactic latitude in the third Fermi-LAT source catalog. None of the unassociated point sources with spectra consistent with PBH evaporation show significant proper motion. Using the nondetection of PBH candidates, we derive a 99% confidence limit on the PBH evaporation rate in the vicinity of Earth, <(rho)over dot>(PBH) < 7.2 x 10(3) pc(-3) yr(-1). This limit is similar to the limits obtained with ground-based gamma-ray observatories.