We study a spinless level that hybridizes with a fermionic band and is also coupled via its charge to a dissipative bosonic bath. We consider the general case of a power-law hybridization function γ (ω) ω r, with r≥0, and a bosonic-bath spectral function B (ω) ωs, with s≥-1. For r<1 and max (0,2r-1) <s<1, this Bose-Fermi quantum impurity model features a continuous zero-temperature transition between a delocalized phase, with tunneling between the impurity level and the band, and a localized phase, in which dissipation suppresses tunneling in the low-energy limit. The phase diagram and the critical behavior of the model are elucidated using perturbative and numerical renormalization-group techniques, between which there is excellent agreement in the appropriate regimes. For r=0, this model's critical properties coincide with those of the spin-boson and Ising Bose-Fermi Kondo models, as expected from bosonization.