Having just gone through the process, I can tell you that paper was the way to go.
The online system is very kludgy. It wasn't easy for me to figure out where I was in the workflow, and trying to add attachments (for me it was a divorce decree) just didn't work.
I printed my application and went to SHP headquarters in Baton Rouge to be fingerprinted, and I had my CHP permit in hand in about a week.
Mike
Thanks for that info. What was the issue with the divorce decree? Was it giving an errror or something?