We are interested in solving the unique continuation problem for the wave equation, i.e., we want to reconstruct the solution of the wave equation given its (noised) value in a subset of the computational domain. Homogeneous Dirichlet boundary conditions are imposed, whereas the initial datum is unknown. We discretize this problem using a space-time discontinuous Galerkin method (including hybrid variables in space and in time) and look for the solution corresponding to the saddle-point of a discrete Lagrangian. We establish discrete inf-sup stability and bound the consistency error, leading to a priori estimates on the residual. Our main result proves the convergence of the discrete solution to the exact solution in a shifted energy norm involving weaker Sobolev norms than the standard energy norm for the wave equation. The proof combines the above a priori bound with a conditional stability estimate at the continuous level. Finally, we run numerical simulations to assess the performance of the method in practice. A static condensation procedure is used to eliminate the cell unknowns and reduce the size of the linear system.