Ground-state two-proton (2p) radioactivity is a rare decay mode found in a few very proton-rich isotopes. The 2p decay lifetime and properties of emitted protons carry invaluable information on nuclear structure in the presence of a low-lying proton continuum. The recently measured 2p decay of ^{67}Kr turned out to be unexpectedly fast. Since ^{67}Kr is expected to be a deformed system, we investigate the impact of deformation effects on the 2p radioactivity. We apply the recently developed Gamow coupled-channel framework, which allows for a precise description of three-body systems in the presence of rotational and vibrational couplings. This is the first application of a three-body approach to a two-nucleon decay from a deformed nucleus. We show that deformation couplings significantly increase the 2p decay width of ^{67}Kr; this finding explains the puzzling experimental data. The calculated angular proton-proton correlations reflect a competition between 1p and 2p decay modes in this nucleus.