Phonon-photon coupling in quantum materials is commonly treated as an effective direct interaction, neglecting the crucial role played by electrons. Here we develop a gauge-invariant theory of cavity phonon polaritons that captures electronic screening, dissipation, and hybridization. We show that screening can suppress polariton formation in metals, severely constraining cavity control of phonon-mediated superconductivity. By contrast, interband electronic resonances in time-reversal-symmetric gapped systems can strongly enhance phonon-photon coupling, enabling robust polariton formation down to the monolayer limit despite electronic absorption. We illustrate these effects in NbSe2, hBN, and gapped graphene. Finally, we show that gauge invariance enforces photon decoupling from a critical soft phonon, establishing a general no-go mechanism for photon condensation.