The Snowman is one of the monsters seen on the whiteboard. It is bet on by Communications Department.
Its appearance likely is that of a typical snowman, perhaps with an evil twist. Like say the snowman's twig arms are replaced with chainsaws or he has spider legs.
A snow globe summons it.
It is not seen during the Purge or anywhere else in the film. If he were in the film, he would have appeared vaguely in the purge or elevator scene