Archive Manager captures information at deployment time, not at render time, so the correct answer is "No".
It will capture all the promotions that you publish, and it will capture the promotions that are active for a page whenever you update that page... but not all variations.
What you can do here is use Personas. You can configure Archive Manager to login to your staging site as a given set of identities, and it will then crawl pages using those identities. If you have SmartTarget promotions being triggered for those user profiles, then the result will be capture in the Archive.
For instance, you could have a promotion for "All visitors living in New York". Set up an identity with a profile stating that the visitor lives in New York, and presto, archive will now capture pages as they look for a user that lives in New York.
As far as I know, the only real way to capture what you show with SmartTarget is via analytics data (making sure you capture TCM IDs in your analytics), custom log parsers, or screen captures (which will be complete overkill on a busy site).
From a legal point of view, all you need to do is to show that you know which promotions may have been shown to a given user, so perhaps you don't really need to capture everything?