I think its kosher to use the prompt for inspiration but I would NOT post or submit it to the LJ once the time allotment has been met. This is a courtesy to the people who run the challenges as well as those who participate by the due dates. But if you wanted to use a prompt for inspiration I'm sure that would be fine, just give a credit source for the prompt - another courtesy. (if you'd like help with wording this, shoot me a PM and I'll do what I can to assist you with something appropriate!)
forthrightly - lovingly called, "forthy" by most - runs the LJ so I suggest getting into contact with her about it if you have any questions about what the rules are for their contests.