The Dark Matter Distributions in Low-Mass Disk Galaxies. I. H$\alpha$
  Observations Using the Palomar Cosmic Web Imager
  Dark-matter-only simulations predict that dark matter halos have cusp-like
inner density profiles, while observations of low-mass galaxies have found a
range of inner slopes that are typically much shallower. It is still not well
established whether this discrepancy can be explained by baryonic feedback or
if it may require modified dark matter models. To better understand the
diversity of dark matter profiles in dwarf galaxies, we undertook a survey of
26 low-mass galaxies ($\log M_*/\textrm{M}_\odot = 8.4-9.8$, $v_{\rm max} =
50-140$ km s$^{-1}$) within 30 Mpc using the Palomar Cosmic Web Imager, which
is among the largest integral field spectroscopic surveys of its type. In this
paper, we derive H$\alpha$ velocity fields for the full sample with a typical
spatial resolution of $\sim$160 pc. We extract rotation curves and verify their
robustness to several choices in the analysis. We present a method for
improving the velocity precision obtained from image slicing spectrographs
using narrowband H$\alpha$ images. For 11 galaxies, we compare the H$\alpha$
velocity fields to CO kinematics measured using CARMA, finding the maps to be
in good agreement. The standard deviation of the difference is typically
$\sim$7 km s$^{-1}$, comparable to the level of turbulence in the interstellar
medium, showing that the two tracers have substantially the same bulk
kinematics. In a companion paper, we will use the rotation curves produced here
to construct mass models of the galaxies and determine their dark matter
density profiles.
