We're a relatively small company and have worked with BRC/HACCP since 2012.
I'm the person in charge of the documentation and two years ago I felt it was time to minimize our paper piles.
It took about six months to do the convertion and I did it so I had about eight months before next BRC audit.
I felt I was well prepared during the audit but had the old folders ready if needed, but it wasn't neccessary.
We're not completely paper free. Production and cleaning documentation has not been converted electronic. Both due to the factory environment is not desirable (wet, humid and cold) and also, some of the staff are not at all willling to even try using other devices except pen and paper.
Maybe it's something we can look into when they've retired... 
After production, the filled in documents are scanned so we can access them electronically.
I looked at different ready-made-systems but the quotes I got were ridiculous and to make changes for it to fit us would make it even more expensive, so in the end we went with Sharepoint.
Since it's already part of our Office-subscription our financial department liked the idea of no additional expenses. 
It doesn't fulfill all needs but almost, so I see it as a good start. Also, all employees now have access to documents etc, wherever they are (as long as they have internet connection).
I download all documents as well so I have access even when the internet's not working (it's been useful the two or three times we've been connectionless these past two years).
At the start, the system consisted of some 6000 files. Don't know if this is a big number or not, but for us it certainly is!
It's a weird feeling, not to have so much paper lying around anymore!