Yes, I was going to write something similar, but the organization will not inform an individual's contact about emergencies that do not involve that individual.
In the event of an emergency, we will contact your nominated "Emergency Contact" immediately.
I realise that probably can't be described as brief, but I would have thought that given the context, it's important to get all the relevant information across, no matter how many words it takes.
For future reference, please note this extract from the Posting Guidelines:
'Thread titles should include all or part of the word/phrase being discussed.' Emergency contact would have been a better (and briefer) title for this thread.