How to get a 3rd party app access to a shared mailbox, or should we convert the mailbox

First, there are a few steps you can take to troubleshoot the issue:

1. Make sure you have the necessary permissions to grant access to the shared mailbox.

2. Verify that the email address of the third-party app is entered correctly.

3. Check the Microsoft -Also the App's- terms of service to ensure that the third-party app is authorized to access Exchange Online.

 In some cases, it may be more appropriate to convert the shared mailbox to a user mailbox and give the third-party app access to that mailbox instead. This can be done by following a similar process as above, but selecting "Convert to regular mailbox" instead of "Add" in the "Permissions" tab. This can be a more secure option, as it allows you to control access to the mailbox through the standard user permissions system, rather than relying on a third-party app's access controls.

Resource: How to Allow a Third-Party App Access to Your Shared Mailbox in Microsoft Exchange – SolYoutionz