How do I add existing Mac Computers to my Apple School Manager without an iPhone?

You can't. The only option (and this is new) is to use Apple Configurator for iPhone.

There is a 'manual instruction' to add Mac or iPhone, within link supplied.

This has a few pre-conditions. ~ Yet, not sure what else may be available.

• Use Automated Device Enrollment - "Add devices Manually" - Apple Support

https://support.apple.com/en-us/HT204142#manual

Add devices manually:

"You can add devices that you didn’t purchase to Automated Device Enrollment, like a donated Mac or iPad. 

..Learn how to manually add iOS devices in Apple Configurator 2 or how to manually add Mac computers in Apple Configurator for iPhone.

..After you manually add a device, assign the device to an MDM server in Apple Business Manager or assign the device to an MDM server in Apple School Manager."

Hey there, and welcome to Apple Support Communities, Andrew_UofU!

From our understanding, you need to add a device to your Apple School Manager. We can certainly understand your concern and can help provide some insight.

We see you recently reviewed the article “Use automated device enrollment”. This is the best article to help add a device.

If you have followed the steps in that article, the next step would be to reach out to Apple Support here: Contact - Official Apple Support

Kindest regards!