In fluids, the compressibility and related thermodynamic properties can be obtained from Kirkwood-Buff integrals (KBIs), i.e., infinite volume integrals over the radial distribution function (RDF). KBI theory has never been applied to crystals because the KBIs diverge when computed in the standard way as running integrals. Here, we show that KBI theory can be applied to solids without divergence, provided that the recently developed finite volume KBI method is used. In order to accelerate the integral convergence as a function of system size, we introduce a physically motivated convolution of the RDF. When using the convoluted RDF and an extrapolation of the finite-volume KBI, the zero-temperature KBI converges very fast to the exact value. We apply the theory to solid argon at finite temperatures. The RDF is computed with a Lennard-Jones potential using Monte Carlo and molecular dynamics simulations, and the isothermal compressibility κT is obtained from the KBI. The variation of κT with temperature agrees very well with experiment. The absolute value of κT is, however, underestimated by 40%-50%, which is attributed to finite size effects of the RDF obtained from molecular simulation. The error can be corrected by a single scaling factor that can be easily calculated at zero temperature. By extending Kirkwood-Buff solution theory to solids, this work lays a new framework for the thermodynamic modeling of complex structures, alloys, and solid solutions.