Hi Elizabeth_2019,
I go through your post carefully and I do understand the inconvenience it has made and apologize for it.
As per your description, it seems external users get error message when they try to access shared SharePoint online library and however, they have granted access to a shared library.
As you mentioned, “We have tried Edge, Chrome and IE and they work on and off” try users change another internet connect and see if there is any difference.
To narrow down the issue, please provide the information below:
1. If you create a new document library and shared libraries with external users, does the issue happen?
2. Does the issue happen to all external users?
I appreciate your patience and understanding.
Best Regards
Waqas Muhammad