how to require users to add file to objects

First of all you should really consider changing Customer Requests to be a document class rather than using a separate object type for it. While it is possible to add files to other object types you will miss out on a lot of functionality tied up especially to the Document object type.
Using a specific document class allows you to add all the properties needed for your specific use case as well as any workflow activities needed to handle the customer request.

With that in place the process would simply be to add the email to that particular document class via drag & drop, copy - paste or even as an automated process where M-Files can retrieve all incoming emails from a designated mailbox.
The latter would be my preferred choice. It is something we use very much for different purposes where information is received via email. If the email has files attached to it you can choose whether such files should remain inside the MSG file or you want M-Files to create a multifile document where the attachments are saved as individual files. If required you can even set up a workflow to separate attached files and add them to individual documents each with their own metadata and workflow. The options for automation are endless!

I've used the Mail Sources before and that is a very nice tool.  I've got development plans to redo this entire setup, but in the short term, is there a way to make this a requirement?

You can via WF State, in Conditions/Files/Contains files = Yes

BR