The intention of the public HIBP domain subscription service is for it to be used by an organisation to monitor their own domains. As such, verification of control is required for every domain they wish to search either by receiving an email to one of several predefined address on that domain (for example, security@), by adding a DNS record, uploading a text file or adding a meta tag.
If you are are responsible for providing services to another organisation which include the need to monitor their domains for breach exposure, you will still need to verify control of any domains to be queried. In some cases you may already have the necessary access (for example, you provide web hosting services and can upload a file or add a meta tag to their website), in other cases you may need your customer's support in order to do this (for example, to forward the verification email to you).
Separately, HIBP provides an Enterprise channel with different services and commercial terms intended for large enterprises and MSPs. The Enterprise channel includes bespoke documentation, contractual terms and other formalities where required. You can submit a request for more information, please include details relating to your organisation and an overview of how you wish to use the service.