In recent years, ultrafaint dwarf (UFD) galaxies have been found through systematic searches of large optical surveys. However, the existence of Leo T, a nearby gas-rich dwarf, suggests that there could be other nearby UFDs that are optically obscured but have gas detectable at nonoptical wavelengths. With this in mind, we perform a search of the full Galactic Arecibo L-band Feed Array HI (GALFA-HI) survey, a radio survey that covers one-third of the sky at velocities -650<V_LSR_<+650km/s, for neutral hydrogen sources. We are able to probe regions of the sky at lower Galactic latitudes and smaller |V_LSR_| compared to previous explorations. We use the Source Finding Application on GALFA-HI and select all sources with similar properties to Leo T and other local dwarf galaxies. We find 690 dwarf galaxy candidates, one of which is particularly promising and likely a new galaxy near the Galactic plane (b=-8{deg}) that is comparable in velocity width and HI-flux to other recently discovered local volume galaxies. We find we are sensitive to Leo T-like objects out to 1Mpc at velocities clear from background HI emission. We check each candidate's corresponding optical fields from Pan-STARRS and fit stars drawn from isochrones, but find no evidence of stellar populations. We thus find no other Leo T-like dwarfs within 500 kpc of the Milky Way in the one-third of the sky covered by the GALFA-HI footprint, and discuss our nondetection in a cosmological context.