The site has a mix of modules installed manually and via Composer: Can I move all to Composer? I've inherited a Drupal 8 site where only few modules were installed via Composer; the rest was manually installed.

Is there a way to fix this, and move the manually installed modules to be handled by Composer?
You can just add the modules to Composer as you would normally: composer require drupal/MODULENAME.

You can then confirm everything is working OK by making a new copy of your site with Composer and verifying that the site loads correctly.

Potential issue:
If you are looking at the list of installed modules to decide what to install, you don't necessarily need to add every module; some may be dependencies of others (for example, with Composer, if you are using Pathauto module, you don't need to install Token explicitly; it will automatically be installed with Pathauto.)  However, it won't hurt anything if you force some dependencies to be installed as well, so this isn't really anything to worry about.