Their faces are dripping water, because some of the snow has started to melt. The writer says "dripping snow" instead of "dripping water" to make it clear that the water that is dripping is melted snow (not sweat, for example). It would sound strange to say "dripping water" in this context, because the reader would wonder why the characters had water on their faces. We can understand why they have snow on their faces, and we know that when snow starts to melt, it will drip.
It's definitely not a common phrase, but anything liquid can "drip," and snow can be partly liquid, depending on how cold it is.