  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 AGN" using the VLA Faint Images of the Radio
Sky at Twenty centimeters (FIRST) radio survey in conjunction with 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-energydistributions (SEDs)
derived from the optical (SDSS) + near-IR (NIR) photometry obtained by us at
the Apache Point Observatory 3.5m. 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 9
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.
