Recent Galactic plane surveys of dust continuum emission at long wavelengths have identified a population of dense, massive clumps with no evidence for ongoing star formation. These massive starless clump candidates are excellent sites to search for the initial phases of massive star formation before the feedback from massive star formation affects the clump. In this study, we search for the spectroscopic signature of inflowing gas toward starless clumps, some of which are massive enough to form a massive star. We observed 101 starless clump candidates identified in the Bolocam Galactic Plane Survey (BGPS) in HCO^+^J=1-0 using the 12m Arizona Radio Observatory telescope. We find a small blue excess of E=(N_blue_-N_red_)/N_total_=0.03 for the complete survey. We identified six clumps that are good candidates for inflow motion and used a radiative transfer model to calculate mass inflow rates that range from 500 to 2000M_{sun}_/Myr. If the observed line profiles are indeed due to large-scale inflow motions, then these clumps will typically double their mass on a freefall time. Our survey finds that massive BGPS starless clump candidates with inflow signatures in HCO^+^J=1-0 are rare throughout our Galaxy.