I would have one partner create a separate page/contact form specifically for the advertiser – so only people who see that contact form are people who were referred to by the advertiser. The advertiser would use that page as their outbound link. I know you can track outbound clicks in Google Analytics events and Contact Form conversions (usually through most contact form plugins) but that is the best way I think. Never done it, but this is how I see most affiliate programs like that work.
The problem with affiliate marketing, like many other home business options, are the so-called gurus and get-rich-quick programs that suggest affiliate marketing can be done fast and with little effort. Odds are you've read claims of affiliate marketing programs that say you can make hundreds of thousands of dollars a month doing almost nothing ("Three clicks to rich!"). Or, they suggest you can set up your affiliate site, and then forget it, except to check your bank deposits.