Partner Portal Lead Form

“Not working well” is a bit vague — better to detail your requirements and how the SFDC form is failing to meet them.

But broadly speaking, you can create a Marketo Referral Form that accepts new leads, noting the original referrer on the referral lead (in a custom field) and also noting the most recent referral’s email address on the referrer.

Well, you can always embed Marketo forms on the sites you have access to, which might not be true for the partner site, as those aren't technically under your control. You can instead get a list of people from the partner in the form of a CSV file and import it into Marketo. You can do a list import in Marketo with the appropriate source, channel, partner, etc. fields included for the right attribution.