The detection of a gamma-ray burst (GRB) in the solar neighborhood would have very important implications for GRB phenomenology. The leading theories for cosmological GRBs would not be able to explain such events. The final bursts of evaporating primordial black holes (PBHs), however, would be a natural explanation for local GRBs. We present a novel technique that can constrain the distance to GRBs using detections from widely separated, non-imaging spacecraft. This method can determine the actual distance to the burst if it is local. We applied this method to constrain distances to a sample of 36 short-duration GRBs detected by the Interplanetary Network (IPN) that show observational properties that are expected from PBH evaporations. These bursts have minimum possible distances in the 10(13)-10(18) cm (7-10(5) au) range, which are consistent with the expected PBH energetics and with a possible origin in the solar neighborhood, although none of the bursts can be unambiguously demonstrated to be local. Assuming that these bursts are real PBH events, we estimate lower limits on the PBH burst evaporation rate in the solar neighborhood.