Recent theoretical studies of topologically nontrivial electronic states in Kondo insulators have pointed to the importance of spin-orbit coupling (SOC) for stabilizing these states. However, systematic experimental studies that tune the SOC parameter λ_{SOC} in Kondo insulators remain elusive. The main reason is that variations of (chemical) pressure or doping strongly influence the Kondo coupling J_{K} and the chemical potential μ-both essential parameters determining the ground state of the material-and thus possible λ_{SOC} tuning effects have remained unnoticed. Here, we present the successful growth of the substitution series Ce_{3}Bi_{4}(Pt_{1-x}Pd_{x})_{3} (0≤x≤1) of the archetypal (noncentrosymmetric) Kondo insulator Ce_{3}Bi_{4}Pt_{3}. The Pt-Pd substitution is isostructural, isoelectronic, and isosize, and it therefore is likely to leave J_{K} and μ essentially unchanged. By contrast, the large mass difference between the 5d element Pt and the 4d element Pd leads to a large difference in λ_{SOC}, which thus is the dominating tuning parameter in the series. Surprisingly, with increasing x (decreasing λ_{SOC}), we observe a Kondo insulator to semimetal transition, demonstrating an unprecedented drastic influence of the SOC. The fully substituted end compound Ce_{3}Bi_{4}Pd_{3} shows thermodynamic signatures of a recently predicted Weyl-Kondo semimetal.