An amplitude analysis of $\Lambda^+_c \to pK^-\pi^+$ decays together with a measurement of the $\Lambda^+_c$ polarization vector in semileptonic beauty hadron decays is presented. A sample of $400\,000$ candidates is selected from proton-proton collisions recorded by the LHCb detector at a center-of-mass energy of 13 TeV. An amplitude model is developed and the resonance fractions as well as two- and three-body decay parameters are reported. The mass and width of the $\Lambda(2000)$ state are also determined. A significant $\Lambda^+_c$ polarization is found. A large sensitivity of the $\Lambda^+_c \to pK^-\pi^+$ decay to the polarization is seen, making the amplitude model suitable for $\Lambda^+_c$ polarization measurements in other systems.