We present Chameleon-Graviton-Condensate Cosmology (CGCC), an effective field theory framework that addresses late-time cosmic acceleration, the $H_0$ tension, and cosmic structure constraints without relying on a conventional cosmological constant. The model posits that high-energy gravitational processes during early cosmic epochs (such as supermassive black hole accretion) inject Kaluza-Klein (KK) gravitons into a higher-dimensional bulk. Following adiabatic cooling and cascade decays in the bulk, these modes re-enter the 3+1 spacetime via a density-dependent chameleon mechanism. Within low-density cosmic voids, where environmental screening is lifted ($m_{phi} to 0$), the re-entering gravitons undergo Bose-Einstein condensation (BEC) into a macroscopic infrared ground state. A quartic self-interaction term stabilizes the condensate, yielding an isotropic equation of state with negative pressure ($w=-1$) that dynamically drives void-dominated expansion. We demonstrate that local void expansion reproduces measurements of $H_0 approx 73text{ km/s/Mpc}$, while spatial integration across void-filament boundaries converges to the global CMB-constrained value of $bar{H}_0 approx 67.4text{ km/s/Mpc}$. Furthermore, the model satisfies strict local constraints from Solar System tests (Cassini bound) and Big Bang Nucleosynthesis (BBN), and offers distinct, falsifiable signatures in weak-lensing shear profiles testable by upcoming astronomical surveys.