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.
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Always disclose your affiliate relationship. Most visitors will probably understand that graphic ads will lead to your getting paid, but if you write a review or use an in-text link as a recommendation, you want your readers to know that may lead to compensation as well. This ensures you retain transparency and trust with your readers, but also, it's required by the FTC's endorsement rules.