You can do it yourself and probably should do it yourself to save money unless you have the kind of site that rarely needs changing, i.e. pure company information, in which case it would not cost you a fortune to let the web design company give your site an annual check and overhaul.

If the web site is completely HTML based, you would be well advised to learn at least the basics of HTML to prevent the site from deteriorating.

If the contents of the site are kept in a database, which is either constantly connected or connected when updates are made, the maintenance becomes much easier as the content change will not necessarily need new HTML coding if it is properly set up from the start. Connecting a site to a database will however mean that the initial cost will be much higher than a pure HTML site.