There are a few ways to track things automatically, but they get complicated:
- referral program software
- Give your referrers special URL's with parameters that identify them as the referrer (like http://url.com/?referrer=JohnDoe), then push that value into a hidden form field
- Create a separate landing page for each referrer

I'd keep things much simpler to start. Just tell your social club that there's a referral program in place, then add a form field on your signup process asking who referred the new customer. If John Doe knows that there's a program in place, when he refers someone, he's likely to tell them "make sure you say I referred you". When the new customer joins, they'll likely remember to enter "John Doe" as the referrer.

It's not bullet proof, but it's an easy way to start.