It's never stated, but it seems likely that the challenge needs to be issued at lupanar for the magic of the ritual to work. The challenge involves cutting a wound to Ulfric's chest, then lupa's own chest (presumably both so that blood is drawn with the wound over heart), and saying the ritualistic words:
The knife is then thrown into the ground so that it's stuck there hilt up. This releases a wave of power that causes arousal and pheromones in lupa, and attracts all the suitable members of the pack to a point where they are all almost entirely beyond reason. The ritual is completed with words:
Afterwards Ulfric has to race and/or fight the rest of the pack for who gets to reach and have sex with lupa first. Losing human shape will drop a contender out of the run.
It's unclear what will happen if Ulfric doesn't win the contest, beyond sex between lupa and someone who is not Ulfric. It would presumably be at the very least a severe hit against Ulfric's authority and might encourage challengers for their leadership.
Known Freyjas Edit
In Blue Moon this role is actually spelled Frejya. However, while Freyja indeed has many variant spellings, in Prose Edda Frejya is listed as Freyja's father, and as such sounds less suitable for this purpose than Freyja herself. It seems likely that the spelling is accidental.