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It is known that Primordial Black Holes (PBHs) can leave an imprint on Cosmic Microwave Background (CMB) anisotropy power spectra, due to their accretion-powered injection of energy into the recombining plasma. Here we study a qualitatively new CMB observable sourced by accreting PBHs: the temperature trispectrum or connected 4-point function. This non-Gaussian signature is due to the strong spatial modulation of the PBH accretion luminosity, thus ionization perturbations, by large-scale supersonic relative velocities between PBHs and the accreted baryons. We first derive a factorizable quadratic transfer function for free-electron fraction inhomogeneities induced by accreting PBHs. We then compute the perturbation to the CMB temperature anisotropy due to a general modification of recombination, and apply our results to accreting PBHs. We calculate a new contribution to the temperature power spectrum due to the spatial fluctuations of the ionization perturbation induced by accreting PBHs, going beyond past studies which only accounted for its homogeneous part. While these contributions are formally comparable, we find the new part to be subdominant, due to the poor correlation of the perturbed temperature field with the standard CMB anisotropy. For the first time, we compute the temperature trispectrum due to accreting PBHs. This trispectrum is weakly correlated with the local-type primordial non-Gaussianity trispectrum, hence constraints on the latter do not lead to competitive bounds on accreting PBHs. We also forecast Planck's sensitivity to the temperature trispectrum sourced by accreting PBHs. Excitingly, we find it to be more sensitive to PBHs under $\sim 10^3 M_{\odot}$ than current temperature-only power spectrum constraints. This result motivates our future work extending this study to temperature and polarization trispectra induced by inhomogeneously-accreting PBHs.