The strong coupling between spin, lattice and electronic degrees of freedom in magnetic materials can produce interesting phenomena, including multiferroic and magnetodielectric (MD) behavior, and exotic coupled excitations, such as electromagnons. We present a temperature- and magnetic-field-dependent inelastic light (Raman) scattering study that reveals the emergence of vibronic modes, i.e., coupled vibrational and crystal-electric-field (CEF) electronic excitations, in the unconventional rare-earth MD material, $text{Ce}_2text{O}_3$. The energies and intensities of these emergent vibronic modes are indicative of enhanced vibronic coupling and increased modulation of the dielectric susceptibility in the Neel state ($T_text{N} approx 6.2,text{K}$). The field-dependences of the energies and intensities of these vibronic modes are consistent with a decrease of both the vibronic coupling and the dielectric fluctuations associated with these modes below $T_text{N}$. These results suggest a distinctive mechanism for MD behavior in $text{Ce}_2text{O}_3$ that is associated with a field-tunable coupling between CEF and phonon states.