Here’s what GRRM has to say:
And, by the way, does it snow in the South during the winter?
Yes, some times, in some places. The Mountains of the Moon get quite a lot of snow, the Vale and the riverlands and the west rather less, but some. King’s Landing gets snow infrequently, the Storm Lands and the Reach rarely, Oldtown and Dorne almost never.
So the Reach rarely has snow even during the winter.