Ok - I've worked it out.
Although I'm the team agent, my developer account is considered to be for an individual (rather than an organisation) and Apple are only giving permission to organisations at present.
The message shown after you apply is confusing, though. It simply says:
<person> requested access to the App Store Connect API on <date>
with no indication that the feature hasn't been rolled out to individuals yet...
I'm also a team member of another account which is for an organisation and, once the team agent had requested access, this page
https://appstoreconnect.apple.com/access/api
changed shortly afterwards to show this message:
Your request to access the App Store Connect API was approved.
and a button below it labelled "Generate API Key"....