We design and analyze an arbitrary-order hybridized discontinuous Galerkin method to approximate the unique continuation problem subject to the Helmholtz equation. The method is analyzed using conditional stability estimates for the continuous problem, leading to error estimates in norms over interior subdomains of the computational domain. The convergence order reflects the Hölder continuity of the conditional stability estimates and the approximation properties of the finite element space for sufficiently smooth solutions. Under a certain convexity condition, the constant in the estimates is independent of the frequency. Moreover, certain weighted averages of the error are shown to converge independently of the stability properties of the continuous problem. Numerical examples illustrate the performances of the method with respect to the degree of ill-posedness of the problem, increasing polynomial order and perturbations in the data.