I'm in a non-Greek fraternal organization where we still have paper applications we fill in by hand, however membership is highly skewed toward senior citizen age so it won't go away for a long time. Our newsletter/magazine is also mail only and is not online.
If electronic submission isn't possible, having forms in Word or as a PDF one can fill out, save, and print would be ideal. I love that Alpha Gam has made our recruitment forms available online so I can access one elsewhere and type in information. With my smartphone I can have one and make sure I get the information I need and put it on the form which I can either print out or email to myself.
My only issue with paperless is security. For some organizations forms could end up on personal computers, a hard drive that is passed around, a college member computer, or somewhere it can be hacked. If there is enough information it could be an issue for identity theft and if bank accounts or credit cards are listed, fraud. With full government names, DOBs, addresses, family names or addresses for legacies (home address as opposed to school), SSN if that group uses it, As long as information can be guaranteed secure, why not. However even paper forms not kept safe or shredded can be at the same risk, so Gusteau better get to shredding