A family of Microsoft products that enable users to capture, organize, and reuse notes electronically.
Dear Samantha,
Good day!!
Per my test, if you are sharing the OneNote Notebook to other users who are part of your organization, the users with at least edit permission will have option to edit the shared Notebook in the OneNote application.
However, if you are sharing the Notebook with Anyone link, the internal users or external users who are part of the organization will not have option to edit the Notebook in the OneNote application. If you have given access to the Notebook with the Full Access permission, then should see the Open in the Desktop app:
Since you have already shared the Notebook with the Full Access permissions, but user is not able to see the Open in Desktop app under Editing, please ask the user to access their Office 365 portal in the InPrivate browser and access the OneNote Notebook link to see whether they have Open in Desktop app under the Editing.
If still the same issue, can you please check the same behavior with the different Notebook and see the outcome?
If the problem still persists, please feel free to post back with us where we will try our best to assist you further.
Thanks for your patience and understanding!!
Best Regards,