A search for charginos and neutralinos at the Large Hadron Collider is reported using fully hadronic final states and missing transverse momentum. 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 background rejection by identifying the high-$pT$ bosons using large-radius jets and jet substructure information. An integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ 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. The 95% confidence level exclusion limits are set on wino or higgsino production with varying assumptions in the decay branching ratios and the type of the lightest supersymmetric particle. A wino (higgsino) mass up to 1060 (900) GeV is excluded when the lightest SUSY particle mass is below 400 (240) GeV and the mass splitting is larger than 400 (450) GeV. The sensitivity to high-mass wino and higgsino is significantly extended compared with the previous LHC searches using the other final states.