Is there a hosting service to make this software an LMS?

Like most eLearning authoring tools Adobe Captivate is meant to collaborate with a Learning Management System. Some of them are free (with limited capabiities) like Moodle.  An alternative whcih exists since many versions Captivate is Internal server reporting. However that means that you'll better hire a programmer to have the minimum functionality of a LMS. That feature (internal server reporting) is the reason for the existence of the two user variables cpQuizInfoStudentName and cpQuizInfoStudentID which areated aatomatically in each project.

The software itself is not an LMS. There is no software or service to make it an LMS. They are separate things.

However, content that is created with Captivate can be delivered through an LMS.

There are many, many different LMS solutions that can be used.

Adobe offers one such solution called Captivate Prime.

You can find out more about it using the link below.

Captivate Prime

I hope this is helpful.