I need to extract the whole information included in Pardot Campaigns and elements inside them like prospects actions and any other activity results. The user is synced to Salesforce, so it seems that the API is not working fine in this case. Is there a connector or a way to grant this kind of reports somehow?
To access the Pardot API, you do need to have a Pardot ONLY user account.
To get the data you are looking for, you will need to look at the Visitor Activity APIs. Depending on how you are looking to work with the data, you might be able to use the "old" API that gives you a bit more granular filtering (but then you need to deal with paging results), or you can use the Bulk Export API (which is a bit different than the other APIs) and filter the results on your side.