We study a recently introduced and exactly solvable mean-field model for the density of vibrational states D(ω) of a structurally disordered system. The model is formulated as a collection of disordered anharmonic oscillators, with random stiffness κ drawn from a distribution p(κ), subjected to a constant field h and interacting bilinearly with a coupling of strength J. We investigate the vibrational properties of its ground state at zero temperature. When p(κ) is gapped, the emergent D(ω) is also gapped, for small J. Upon increasing J, the gap vanishes on a critical line in the (h, J) phase diagram, whereupon replica symmetry is broken. At small h, the form of this pseudogap is quadratic, D(ω) ∼ ω$^2$ , and its modes are delocalized, as expected from previously investigated mean-field spin glass models. However, we determine that for large enough h, a quartic pseudogap D(ω) ∼ ω$^4$ , populated by localized modes, emerges, the two regimes being separated by a special point on the critical line. We thus uncover that mean-field disordered systems can generically display both a quadratic-delocalized and a quartic-localized spectrum at the glass transition.