Experimental evidence is presented of an increase in the relaxation accompanying hole emission from the vacancy in unstrained p-type Si1-xGex alloys as compared with pure silicon. The vacancy is known to destroy the rigidity of the lattice which in turn strongly affects the vibrational modes, making large lattice relaxation possible. As the Ge content increases, our deep-level transient spectroscopy measurements reveal a constant hole ionization enthalpy with a dramatic increase of the emission rate prefactor. The analysis shows that this increase can be accounted for by an entropy term caused by a lattice relaxation specific to the statistical alloy disorder and responsible for a large carrier-phonon coupling around the vacancy.