Primordial black holes (PBH) with masses below approximately $10^{15}$ g are expected to emit gamma rays with energies above a few tens of MeV, which are capable of being detected by the Fermi Large Area Telescope (LAT). 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 case of individual PBHs, the Fermi LAT is most sensitive to PBHs with temperatures near 16 GeV, which it can detect out to a distance of about 0.03 pc. These PBHs would appear as potentially moving point sources. We develop a new algorithm to detect the proper motion of a gamma-ray PS, and apply it to unassociated PS in the third Fermi-LAT source catalog (3FGL). None of unassociated PS with spectra consistent with PBH evaporation show significant proper motion. The derived 99% confidence limit on PBH evaporation rate in the vicinity of the Earth is similar to the limits obtained with ground-based gamma-ray observatories.