@Arthur , Thanks for posting a good question. Apologies for the delay!
Azure DNS doesn't currently support the option to buy domain names. To buy domains, you must use a third-party domain name registrar.
The good news is that the feature to buy domain names is tracked in the Azure backlog. Use the feedback site to register your support for this feature: Azure should be its own domain registrar - As mentioned by our product engineering team in the post "Domain registration is being evaluated and remains on our long term road-map"
If your requirement fits, you can leverage delegation of DNS zones with Azure DNS
Azure DNS allows you to host a DNS zone and manage the DNS records for a domain in Azure. In order for DNS queries for a domain to reach Azure DNS, the domain has to be delegated to Azure DNS from the parent domain. This article explains how domain delegation works and how to delegate domains to Azure DNS.