When the domain name is first registered, your organization should be the owner of the domain name. Changing the owner of a domain name involves a "transfer" form that must be notarized by both parties, and some additional fees.
As mentioned earlier, the cost of the domain name itself is a charge, and the company that provides the domain name, Internet for example, will bill you directly for the domain name itself.
Recommendations:
Make sure that whoever registers the domain name specifies your name, or the name of your business or organization, as appropriate, as the owner.
