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We study a recently introduced and exactly solvable mean-field model for the density of vibrational states $\mathcal{D}(ω)$ 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 $\mathcal{D}(ω)$ 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, $\mathcal{D}(ω)\simω^2$, 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 $\mathcal{D}(ω)\simω^4$, 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.