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Some of my other sponsors don't allow this, but allow redirection to a thank you page.
Could I still track converting keywords (organic) without adding the tracking pixel on the affiliate sponsor landing pages?
Normally a conversion is tracked by the completion of an event which is why the "thank you" page has the code (or the conversion can be picked up by the redirect to a "thank you" page depending on the anlaytics tools used).
The simple answer to your question though is you need to have a tracking code or tracking call of some kind on an event completion to confirm a conversion for your keywords.