We introduce the Swift Gamma-Ray Burst Host Galaxy Legacy Survey ("SHOALS"), a multi-observatory high-redshift galaxy survey targeting the largest unbiased sample of long-duration gamma-ray burst (GRB) hosts yet assembled (119 in total). We describe the motivations of the survey and the development of our selection criteria, including an assessment of the impact of various observability metrics on the success rate of afterglow-based redshift measurement. We briefly outline our host galaxy observational program, consisting of deep Spitzer/IRAC imaging of every field supplemented by similarly deep, multicolor optical/near-IR photometry, plus spectroscopy of events without preexisting redshifts. Our optimized selection cuts combined with host galaxy follow-up have so far enabled redshift measurements for 110 targets (92%) and placed upper limits on all but one of the remainder. About 20% of GRBs in the sample are heavily dust obscured, and at most 2% originate from $z\gt 5.5.$ Using this sample, we estimate the redshift-dependent GRB rate density, showing it to peak at $z\sim 2.5$ and fall by at least an order of magnitude toward low (z = 0) redshift, while declining more gradually toward high ($z\sim 7$) redshift. This behavior is consistent with a progenitor whose formation efficiency varies modestly over cosmic history. Our survey will permit the most detailed examination to date of the connection between the GRB host population and general star-forming galaxies, directly measure evolution in the host population over cosmic time and discern its causes, and provide new constraints on the fraction of cosmic star formation occurring in undetectable galaxies at all redshifts.