In order to find more examples of the elusive high-redshift molecular absorbers, we have embarked on a systematic discovery program for highly obscured, radio-loud 'invisible active galactic nuclei' using the Very Large Array Faint Images of the Radio Sky at Twenty centimeters radio survey in conjunction with the Sloan Digital Sky Survey (SDSS) to identify 82 strong ({>=}300 mJy) radio sources positionally coincident with late-type, presumably gas-rich galaxies. In this first paper, the basic properties of this sample are described including the selection process and the analysis of the spectral energy distributions (SEDs) derived from the optical (SDSS) + near-IR (NIR) photometry obtained by us at the Apache Point Observatory 3.5 m. The NIR images confirm the late-type galaxy morphologies found by SDSS for these sources in all but a few (6 of 70) cases (12 previously well studied or misclassified sources were culled). Among 70 sources in the final sample, 33 show galaxy type SEDs, 17 have galaxy components to their SEDs, and 20 have quasar power-law continua. At least nine sources with galaxy SEDs have K-band flux densities too faint to be giant ellipticals if placed at their photometric redshifts. Photometric redshifts for this sample are analyzed and found to be too inaccurate for an efficient radio-frequency absorption line search; spectroscopic redshifts are required. A few new spectroscopic redshifts for these sources are presented here but more will be needed to make significant progress in this field. Subsequent papers will describe the radio continuum properties of the sample and the search for redshifted H I 21 cm absorption.