As far as I know there is no mechanism in place for an Org Admin or App Developer to request access to APIs. By default they only have read access to the API Catalog and the API Explorer.
As of today:
- Applications are assigned to an Organization by the Portal Admin
- Applications are assigned APIs by the Portal Admin
- Developers belong to/register with an Organization, with permission/approval from the Portal Admin
- Developers can use the Applications that belong to their Organization to access the assigned APIs
There may be more other information available; if so I defer to Portal admins with more knowledge.
Cheers,
-case-