Understanding how local distortions determine the functional properties of high entropy materials, containing five or more elements at the same crystallographic site, is an open challenge. We address this for a compositionally complex spinel oxide (Mn_0.2Co_0.2Ni_0.2Cu_0.2Zn_0.2)Cr₂O₄ (A⁵Cr₂O₄). By comparatively examining extended X-ray absorption fine structure on A⁵Cr₂O₄ and its parent counterparts, ACr₂O_4, along with density functional theory calculations for multiple configurations, we find that the element-specific distortions go beyond the first neighbor. Specifically, the strong Jahn-Teller distortion present in CuCr₂O₄ is found to be completely suppressed in A⁵Cr₂O₄ even locally. Instead, there is a broad distribution of Cu-O and Cu-Cr bond distances, while other A-O distances acquire certain specific values. This study demonstrates the additional flexibility of a cationic sublattice in maintaining a uniform long-range structure, in contrast to previous reports showing only the accommodative anionic sublattice. The mean-field magnetic interactions of A⁵Cr₂O₄ exhibit a striking resemblance to those of NiCr₂O₄, despite the presence of multiple magnetic ions and variable bond lengths. This originates from the comparability of bond lengths around Cr in both materials. Our study paves the way for a deeper understanding of the impact of local structural distortions on the physical properties of compositionally complex quantum materials.