I don't think such a theorem has been proved for the random Lorentz gas. First I want to point out that Sinai proved those scaling limit results for the case of (2D) periodic Lorentz gas with finite horizon (finite maximum free path for the particle). The case of periodic Lorentz gas with unbounded horizon was studied until recently, and a lot of places reference the work by Szasz and Varju ("Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon"). One gets analogous diffusion results but the scaling factor is different here.
A very recent article which seems to do a good job at surveying these results is "New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis" by C.P. Dettmann, which considers higher dimensional Lorentz gases. There is also this talk with the same title summarising the results, it says there that the periodic case is the only one which has been treated rigorously so far, and that the random Lorentz case is expected to have similar diffusion properties, but we have no proof.
