Ultralight scalars arise naturally in many extensions to the Standard Model and are compelling dark matter candidates. Around spinning black holes, dense scalar clouds could form through the conversion of rotational energy into particles via black hole superradiance. Extreme-mass-ratio inspirals (EMRIs) targeted by future space-based detectors will give us unparalleled access to the environments of massive black holes, allowing us to probe the presence of scalar clouds. We consider EMRIs around a Schwarzschild black hole and show that eccentricity induces a dense sequence of resonances in the scalar fluxes near the last stable orbit. These resonances arise only in a fully relativistic treatment, as they are intrinsically tied to the splitting between the azimuthal and radial orbital frequencies in the strong-field regime. By evolving the orbits adiabatically, we show that the resulting resonant transitions substantially enhance the exchange of energy and angular momentum between the EMRI and the scalar cloud, significantly amplifying the accumulated dephasing in the gravitational waveform relative to circular motion. Our results highlight the importance of eccentricity in shaping the observational signatures of EMRIs embedded in scalar clouds.