I've used the site a bunch of times and have had no complaints.

Having booked through them on LH recently, I know that the record locator FlightNetwork provides in their confirmation is unrelated to airline's booking code. LH will pull up the reservation using the one given by the them, but they are probably one of the exceptions.
I would first first contact FlightNetwork and see if they can give it to you (being the service provider, they should be able to look it up), and if they can't, then contact the airline.