Apparently, all it takes to offer a nomination is to write a letter to the committee. But the committee says they only accept nominations from people who meet their criteria
https://www.nobelpeaceprize.org/nobel-peace-prize/nomination/criteria-for-nominators
"Journalists" are not in that list. But as a former prize winner himself, if Obama were to forward the letter along and say he endorses it, that might be enough to get it over the line.
Crikey