You should always maintain control of your domain name(s). Any convenience achieved by allowing your website manager to own your domain, can be greatly outweighed by the trouble you will have if things goes wrong. It doesn’t really matter how much you trust your own webmaster, no one else should ever have the ultimate control of your domain name. It is extremely easy to find and register a domain name on your own, without any outside help. So there is no reason to letting anyone else control your domain. It’s simply too easy to do — and the downside of not doings so is tremendous.
Some unscrupulous website business may try to appeal to convenience by offering to register your domain name for you. While this may appeal to anyone new to websites, you should never fall for it. Now, I am not saying that anyone doing this is trying to rip off your domain name, but it does happen. I’ve heard of a real stories where the webmaster literally held a domain name for ransom, after securing it for his client. They will often ask for thousands of dollars — for a domain that cost them $20 or less to register, and is rightfully yours. So always register your own domain name if you can. If for any reason you need to rely on a third party, make sure you have some sort of written promise that the domain rightfully belongs to you.
If your website is vital to your business, then so is your domain name. It is a vital aspect of your business that needs to be protected, just like your business name itself. You would never let anyone else register your business name for you. The reasons for this are obvious and self-evident. And so they are for your domain name.
Since it so easy to register a domain, there is literally no reason for anyone to do it but you. There are many registrars to choose from, but I like using Google myself. Anyone who has a gmail account can buy and register domain names. Simply search for “Google Domains” while you are logged into your gmail account. Once there, you can try different domains to see if they are available, and buy one if you like. You can also share them with other people through their gmail accounts, including your webmaster. This way, your webmaster can make any configuration modifications that are needed to your website, email, etc. But you still retain full control of the domain, granting or denying access at will.
So always register your domains yourself. Even if it isn’t Google, there are always ways of allowing access to others if needed without giving up your rights.