  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 which covers one-third of the sky at
velocities $-650 < V_{\rm{LSR}} < +650 \; \rm{km} \; \rm{s^{-1}}$, 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 (SoFiA) 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 ($b=-8^{\circ}$) 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 $1 \; \rm{Mpc}$ 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
$500 \; \rm{kpc}$ 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.
