  We analyze the spectral and transport properties of the interacting
disordered Tavis-Cummings model at half excitation filling. We demonstrate that
a Poissonian level statistics coexists with eigenfunctions that are
multifractal (extended, but non-ergodic) in the Hilbert space, for all
strengths of light-matter interactions. This is associated with a lack of
thermalization for a local perturbation. We find that the bipartite
entanglement entropy grows logarithmically with time, similarly to many-body
localized systems, while the spin imbalance tends to zero for strong coupling,
in analogy to ergodic phases. We show that these effects are due to the
combination of finite interactions and integrability of the model. When a small
integrability-breaking perturbation (nearest-neighbor hopping) is introduced,
typical eigenfunctions become ergodic, seemingly turning the system into a
near-perfect conductor, contrary to the single-excitation noninteracting case.
We propose a realization of this model with cold atoms.
