Error on FastLane (NSF) letter of Reference System for the NSFPF
Screen shot and date stamp every attempt to do this operation until it works. Call the helpdesk 800 number back every hour during business hours until someone answers the phone, but don't just leave message after message if it goes to voicemail. If you haven't already, send invites via email personally to each of your references so that they know the request from FastLane will be coming eventually. Tell each of your references about the problem so that they'll be prepared to take alternative action if NSF gives you a workaround. Keep good records of all your attempts.

If you can figure out who the cognizant program officer is for the specific fellowship you are applying for, write them an email Monday after you have tried to get in touch with the FastLane user support folks again with a short explanation of your problem and ask them for options.

FastLane being down on broken is usually a great reason for NSF to give you an exception to a deadline. 

As noted in the comments, there is some upcoming Fastlane downtime due to the NSF financial system upgrade, but that shouldn't be the problem today or for the rest of the week. The FastLane user support team is usually very responsive, but with the financial system update going on, they may be swamped. 

Finally, you might also try talking to the Sponsored Research Office/Office of Sponsored Projects/etc. at your current university. They work with staff at NSF all the time, and may have contacts that can help you.