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We present multi-wavelength observations of two gap transients followed by the Carnegie Supernova Project-II and supplemented with data obtained by a number of different programs. Here in the first of two papers, we focus on the intermediate luminosity red transient (ILRT) designated SNhunt120, while in a companion paper we examine the luminous red novae AT 2014ej. Our data set for SNhunt120 consists of an early optical discovery, estimated to be within 3 days after outburst, the subsequent optical and near-infrared broadband followup extending over a $\sim$2 month period, two visual- and two near-infrared wavelength spectra, and Spitzer Space Telescope observations extending from early ($+$28 d) to late ($+$1155 d) phases. SNhunt120 resembles other ILRTs such as NGC 300-2008-OT and SN 2008S, and like these other ILRTs, SNhunt120 exhibits prevalent mid-infrared emission at both early and late phases. From the comparison of SNhunt120 and other ILRTs to electron-capture supernova simulations, we find that the current models underestimate the explosion kinetic energy and thereby produce synthetic light curves that over-estimate the luminosity. Finally, examination of pre-outburst Hubble Space Telescope images yields no progenitor detection.