Dan, what sort of group does this non-admin user belong to? You'll need to provide appropriate visibility to the user/user group. Could test with Developers Group (available by default) This group by default have read & update rights. For more you can check the User Group documentation Or this Security fundamental video
Hope this'd help.
Hi Dan Mahoney,
When you will open the Mashup with Non-Admin user; some errors will show in the Application logs regarding the Permissions. Just provide those permissions to the Non-Admin user and it will work fine.
Do let us know in case you face any issue in understanding the logs.
I hope it helps.