The REST API allows you to retrieve DOI registration statistics.

Examples:

Use the following query to retrieve all the clients of e.g Texas Digital Library https://api.datacite.org/clients/totals?provider-id=tdl

The query can be extended to filter out the draft DOIs by specifying the "state" e.g. https://api.datacite.org/clients/totals?provider-id=tdl&state=findable

Read more about setting up queries with the REST API here.