We report near-infrared (0.7-2.5 micron) reflectance spectra for each of the six target asteroids of the forthcoming NASA Discovery-class mission Lucy. Five Jupiter Trojans (the binary (617) Patroclus system, (3548) Eurybates, (21900) Orus, (11351) Leucus, and (15094) Polymele) are well-characterized, with measurable spectral differences. We also report a survey-quality spectrum for main belt asteroid (52246) Donaldjohanson. We measured a continuum of spectral slopes including red (Orus, Leucus), less red (Eurybates, Patroclus-Menoetius) and intermediate (Polymele), indicating a range of compositional end-members or geological histories. We perform radiative transfer modeling of several possible surface compositions. We find that the mild-sloped spectra and low albedo of Patroclus and Eurybates imply similar compositions. Eurybates (~7 wt.% water ice) and Patroclus (~4 wt.% water ice) are consistent with a hydrated surface. Models for Orus and Leucus are consistent with each other and require a significantly more reddening agent (e.g. iron-rich silicates or tholin-like organics). Polymele has a linear spectrum like Patroclus, but a higher albedo more closely aligned with Orus/Leucus, defying simple grouping. Solar system formation models generally predict that the Jovian Trojans accreted in the outer solar system. Our observations and analysis are generally consistent with this expectation, although not uniquely so.