As of August 2023, some domains require a paid subscription to be searched. Most domains fit within the free tier, but some domains require a paid subscription based on the number of email addresses that have appeared in breaches. Paid subscriptions provide the ability to search domains via the web interface, via the domain search API and also provide access to formal support via the portal at support.haveibeenpwned.com.
You can take out a paid subscription directly from the domain search dashboard. Here you can also add domains for monitoring (you must be able to demonstrate that you control the domain first), and see how many breached addresses are on them before you choose a paid subscription.