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Here, we explore component-dependent local lattice distortions in polycrystalline, equiatomic, face-centered cubic CrMnFeCoNi high-entropy alloys and their modifications induced by dilute interstitial carbon. Multi-edge extended X-ray absorption fine structure spectroscopy combined with reverse Monte Carlo analysis reveals that the Cr component experiences the most substantial local distortions, independent of the temperature of prolonged annealing treatments (993 K or 1373 K) and the nominal carbon content (0 to 0.8 at.%). The static disorder around Cr atoms was found to increase markedly and monotonically upon carbon alloying, whereas Mn, Fe, Co, and Ni demonstrate weaker and non-monotonic tendencies. The carbon-induced lattice distortions extend over several coordination shells, indicating the pronounced effect of the carbon presence on the local environment around Cr absorbers. First-principles density functional theory and finite-temperature molecular dynamics simulations confirm the greater impact of carbon on the local lattice distortions around Cr than around the other 3d constituent elements, based on the previous finding that carbon preferentially occupies Cr-rich interstitial sites. These results provide decisive hints towards the atomistic origin of the non-monotonic diffusion behavior previously reported for carbon-doped CrMnFeCoNi alloys, and are noticeable for understanding the carbon-induced phase transitions in compositionally complex systems.