We study the zero-point and thermal ionic motion in La2CuO4 by means of high-resolution neutron-diffraction experiments. Our results demonstrate anisotropic motion of O and, to a lesser extent, Cu ions, both consistent with the structure of coupled CuO6 octahedra, and quantify the relative effects of zero-point and thermal contributions to ionic motion. By substitution of O-18, we find that the oxygen isotope effect on the lattice dimensions is small and negative (-0.01%), while the isotope effect on the ionic displacement parameters is significant (-6 to 50%). We use our results as input for theoretical estimates of the distribution of magnetic interaction parameters, J, in an effective one-band model for the cuprate plane. We find that ionic motion causes only small (1%) effects on the average value < J >, which vary with temperature and O isotope, but results in dramatic (10-20%) fluctuations in J values that are subject to significant (8-12%) isotope effects. We demonstrate that this motional broadening of J can have substantial effects on certain electronic and magnetic properties in cuprates.