We have our clients buy their own domain names so they have ownership of them. Having a domain in your name is as important as having your business in your name.
We've had too many clients come to us where their former website developer owned the domain of the website, which is usually in a single account with all their other domains, and sometimes it's difficult to get ownership. There have been a number of times when that former developer refused to turn over the domain. You do not want to be in this situation, so prevent that, just buy your domain name yourself.
Domain names cost about $10/year. You should register them for the maximum time allowed by the domain name registrar (like 10 years) since that's a signal to search engines like Google that you plan on running this website and business for a long time.