Does Dynamic Objects only work if the other system has an API?

Yes and No. 

Dynamic Objects enables admins to define a string pattern to be used in two ways, one of which does not require API access. 

If you want to simply append the string to the end of a URL, you can do that with Dynamic Objects without API access.

If you want to use that string to look up information from an external system, that system needs to have an open API and the necessary endpoints to support this workflow.