Yes, and no. It is important to understand some details.
For SaaS websites, the site is subscription-based. You will have full access to your SaaS site as long as you continue to pay the subscription. However, these websites cannot be downloaded and taken elsewhere, so technically your website is more "lease" than "own".
For CMS websites, the website sits in a hosting account. For SMB sites we generally recommend purchasing a site via a hosting provider but you can also build and maintain your own hosting if you want to. The website software and database can be downloaded, modified offline and uploaded to a different hosting account if you want to.
If you require "sole custody" of your website once it is launched, we recommend a CMS website, with your own domain name and website hosting account. We can build a website in your account, and once the project is complete you can remove our access and change your passwords.
Further details regarding ownership are available in the Website Development Contract, which is signed prior to the commencement of a project.
Read our CMS v SaaS Platform Comparison.