Previous work in material sensing with soft robots has focused on integrating flexible force sensors or optical waveguides to infer object shape and mass from experimental data. In this work, we present a novel modular sensing platform integrated into a hybrid-manufactured soft robot gripper to collect and process high-fidelity spectral information. The custom design of the gripper is realized using 3D printing and casting. We embed full-spectrum light sources paired with lensed fiber optic cables within an optically clear gel, to collect multi-point spectral reflectivity curves in the Visible to Near Infrared (VNIR) segment of the electromagnetic spectrum. We introduce a processing pipeline to collect, clean, and merge multiple spectral readings. As a demonstration of sensor capabilities, we gather sample readings from several similarly-shaped and textured items to show how spectroscopy enables explainable differentiation between objects. The integration of spectroscopic data presents a promising new sensing modality for soft robots to understand the material composition of grasped items, facilitating numerous applications for food-processing and manufacturing.