We don't really have any way to "interrupt" the transition from cart to checkout, so you couldn't request the referral emails after the buyer clicks the checkout button but before they actually reach checkout.
As best I can figure, you could just have a script in your own page/site that would handle requesting the 3 emails and sending the message to them, and when the buyer enters those emails, that would reveal a discount code they could enter in their cart to get the discount before they click the checkout button.
You may want to create this discount with a list of single-use Discount Codes (Max. Use Count = 1), so you could issue one code to each buyer providing referrals, and once they use it they can't use it again nor share it with others.
