Sometimes. Usually we would discard your existing site and create a new one for you. Our technology allows us to do this quickly and delivers a site that allows faster site maintenance.
When your site was developed then the developers would have used a website building package such as Dreamweaver. The majority of old-school developers use Dreamweaver, so we will use that as an example. (We use our own product called Jasmine, which also provides a Content Management System. In line with many modern companies we provide websites on an SaaS basis, which means that we charge for the service on an ongoing basis, and our service is all-inclusive.)
In the past companies would develop on products such as Dreamweaver and upload your site via FTP to a webspace. To maintain the website you would need to go back to the developers and ask them to make changes. If you want to change the company providing this development support then you would usually need the original Dreamweaver files from the original developers so that you could pass these on to your new developers.
Note that you should never have more than one organisation concurrently making changes using software such as this, because one organisation will overwrite the changes made by the other.
In practice not many companies have ever asked their developers for these files, so a new developer will usually build a new website for you.
We do not support sites written in Dreamweaver. However, in some small websites we can work directly on the HTML of the webpages to make minor alterations. Note that if we do this then our changes will not be reflected in the Dreamweaver source of the website if you also have somebody else making updates to the site, so our changes will be removed then next time the other people make some changes to the site.