We obtain CO(1‑0) molecular gas measurements with the Australia Telescope Compact Array on a sample of 43 spectroscopically confirmed Hα emitters in the Spiderweb protocluster at z = 2.16 and investigate the relation between their star formation activities and cold gas reservoirs as a function of environment. We achieve a CO(1‑0) detection rate of ∼23 ± 12% with ten dual CO(1‑0) and Hα detections within our sample at 10 < log M*/M⊙ < 11.5. In addition, we obtain upper limits for the remaining sources. In terms of total gas fractions (Fgas), we find our sample is divided into two different regimes mediated by a steep transition at log M*/M⊙ ≈ 10.5. Galaxies below that threshold have gas fractions that in some cases are close to unity, indicating that their gas reservoir has been replenished by inflows from the cosmic web. However, objects at log M*/M⊙ > 10.5 display significantly lower gas fractions than their lower stellar mass counterparts and are dominated (12 out of 20) by objects hosting an active galactic nucleus (AGN). Stacking results yield Fgas ≈ 0.55 for massive emitters excluding AGN, and Fgas ≈ 0.35 when examining only AGN candidates. Furthermore, depletion times of our sample show that most Hα emitters at z = 2.16 will become passive by 1 < z < 1.6, concurrently with the surge and dominance of the red sequence in the most massive clusters. Our environmental analyses suggest that galaxies residing in the outskirts of the protocluster have larger molecular-to-stellar mass ratios and lower star formation efficiencies than galaxies residing in the core. However, star formation across the protocluster structure remains consistent with the main sequence, indicating that galaxy evolution is primarily driven by the depletion of the gas reservoir towards the inner regions. We discuss the relative importance of inflow and outflow processes in regulating star formation during the early phases of cluster assembly and conclude that a combination of feedback and overconsumption may be responsible for the rapid cold gas depletion these objects endure.