The employment of periodic arrays of plasmonic nanoparticles has been proposed by several groups for enhanced transmission or absorption and for realizing optical nanowaveguides. Generally, due to their small transverse dimensions, such linear arrays have been operated near their dipolar resonance. However, it has been recently shown that nanoscale plasmonic particles may also support higher-order resonances, which provide some advantages in different applications. Here we derive a full-wave analytical closed-form dispersion equation for the guided and leaky modes supported by linear chains of nanoparticles near a quadrupolar resonance. We show that, despite the vanishing bandwidth of the individual quadrupolar resonance in each of the nanoparticles composing the chain, the overall bandwidth of quadrupolar chain guidance is relatively large due to strong coupling, even considering realistic losses and frequency dispersion of optical materials. Applications for low-damping optical nanotransmission lines and leaky-wave nanoantennas are suggested.