We show that the structure, properties, and concentration of vacancies in crystals can be studied by ultrasonic experiments previously employed for impurity centres only. Measurements of the temperature dependence of attenuation and phase velocities of ultrasonic shear waves of 52MHz propagating along the crystallographic axis [110] of nominally pure ZnSe single crystals (grown by the seeded physical vapour transport method) show strong anomalies which are typical for relaxation processes in system with isolated Jahn-Teller (JT) centres. The observed JT distortion mode is trigonal, subject to a threefold orbitally degenerate T-term interaction with trigonal and tetragonal nuclear displacements. In the absence of sufficiently high concentrations of impurity atoms with such properties we attributed the observed JT centres to zinc vacancies. The temperature dependence of the isothermal and adiabatic forms of the appropriate elastic modulus and the relaxation time show that the relaxation mechanism changes from thermal activation at higher temperatures to tunnelling through a potential energy barrier below 18K. We provide an estimate of the magnitude of the potential barrier, as well as the pseudorotation frequency and concentration of vacancies. Also we determine the extremum points of the adiabatic potential energy surface of the vacancy centre.