Pyrochlore cobalt fluoride NaCaCo₂F₇ is a geometrically frustrated magnet, which exhibits spin freezing below T_f ~ 2.4 K while the Weiss temperature is θ_W ~ -140 K. The spin-freezing behavior in this compound is considered to be due to the presence of bond randomness which is generated by the random occupation of Na⁺ and Ca²⁺ ions on the pyrochlore A sites. We perform ultrasound velocity measurements in a single crystal of NaCaCo₂F₇. Temperature (T) dependence of the bulk modulus C_B = (C₁₁+2C₁₂)/3 exhibits Curie-type -1/T softening upon cooling below ~ 20 K down to T_f, which should be a precursor to the lattice distortion at T_f. And this softening is suppressed by the application of magnetic field, indicating that the spin freezing is driven by the spin-lattice coupling. For the bond-disordered frustrated NaCaCo₂F₇, the softening in the breathing elastic mode of C_B above T_f suggests that the spin freezing, or the release of frustration is a result of the enhancement of the bond randomness driven by the spin-lattice coupling.