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.
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