A search for charginos and neutralinos at the Large Hadron Collider using fully hadronic final states and missing transverse momentum is reported. Pair-produced charginos or neutralinos are explored, each decaying into a high-pT Standard Model weak boson. Fully hadronic final states are studied to exploit the advantage of the large branching ratio, and the efficient rejection of backgrounds by identifying the high-pT bosons using large-radius jets and jet substructure information. An integrated luminosity of 139 fb(-1) of proton-proton collision data collected by the ATLAS detector at a center-of-mass energy of 13 TeV is used. No significant excess is found beyond the Standard Model expectation. Exclusion limits at the 95% confidence level are set on wino or higgsino production with various assumptions about the decay branching ratios and the type of lightest supersymmetric particle. A wino (higgsino) mass up to 1060 (900) GeV is excluded when the lightest supersymmetry particle mass is below 400 (240) GeV and the mass splitting is larger than 400 (450) GeV. The sensitivity to high-mass winos and higgsinos is significantly extended relative to previous LHC searches using other final states.