In the present work, a Lattice Boltzmann formulation in vorticity-stream function variables is proposed for axisymmetric flows with swirl. For this purpose, several source terms are proposed and implemented. Although containing velocity gradients, these sources are in the Lattice Boltzmann framework and fulfill the Euler and Navier-Stokes equations in their conservative form. The main characteristics of the proposed method are: First, the momentum equation is solved using a unified Lattice Boltzmann solver; second, the proposed sources are consistent with the nonviscous and viscous momentum equations; and third, the implemented method is second-order accurate in space. Numerical tests on the Taylor-Couette flow with finite aspect ratio of 3.8 and the lid-driven cylindrical cavity flow were carried out showing good agreement with numerical and experimental results found in the literature, evidencing the ability of the implemented method to solve axisymmetric flows with swirl. In the case of the lid-driven cylindrical cavity flow, the implemented method is able to correctly reproduce some qualitative characteristics of this flow such as the vortex breakdown close to the cavity axis at different Reynolds numbers and cavity aspect ratio.