When the snowman brings the snow…
Let us unpack this. Sort of reverse engineer the snowball, as it were.
A snowman is traditionally a humanoid depiction made from fresh snow. Therefore the snowman is snow. The only snow that the snowman can technically 'bring' is the snow of which he is composed.
However, the snowman also tends to be immobile. So no act of bringing is in fact possible, at least not in the physical sense.
He might, for the sake of argument, 'summon' the snow. By this I mean will the very substance of life, his life at least, into being. In this particular scenario, the snowman is more of a 'snow sage' or a 'winter wizard' if you will. Suffice to say I have yet to come across such a powerful snowman.
That leaves us with a conceptual meaning. The snowman’s mere existence brings the thought of snow into the mind of the observer. This is by far the most straightforward explanation. That being said, it is still specious: unless the observer’s view is somehow limited to only the snowman and not the probably large bed of snow underneath. This would require a 'Plato’s Cave' level of logic.
Nevertheless, I highly doubt that the lyricist strove for that particular degree of realism. Still one would expect more of Roy Wood. If a Christmas song is worth writing then it is worth writing with proper and thoroughgoing attention to detail.
Incidentally, Merry Christmas Everyday, I suppose…