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 case of individual PBHs, the Fermi LAT is most sensitive to PBHs with temperatures above approximately 16 GeV and masses $6times 10^{11}$ 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 becomes 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 a gamma-ray point sources, and apply it to 318 unassociated point sources at high galactic latitude in the third Fermi-LAT source catalog (3FGL). None of unassociated point sources with spectra consistent with PBH evaporation show significant proper motion. Using the non-detection of PBH candidates, we derive a 99% confidence limit on PBH evaporation rate in the vicinity of the Earth $dot{rho}_{rm PBH} < 7.2 times 10^3: {rm {pc}^{-3} {yr}^{-1}}$. This limit is similar to the limits obtained with ground-based gamma-ray observatories.
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