Due to a Microsoft limitation, it's currently not possible to grant permission to read an entire mailbox. The “ReadPermission” access right refers to the ability to read the permission on the mailbox, but it doesn't grant any other permissions to the mailbox, including “Read-only”. However, you can grant “Read and manage delegation”, which is equivalent to “Full permission”. This allows a delegate to open the mailbox and act as the mailbox owner.
Read permission is only applicable for on-premise use. It's not intended for end-user access.
To check other mailbox permissions, please refer to the article How to grant access to mailbox.
However, you can grant "Read Permission' at the folder level. This can be done using PowerShell or by using a custom action in CoreView.
Mailbox folder permission
PowerShell cmdlet for Mailbox Folder level permission:
To add folder-level permissions for users in mailboxes you can use PowerShell cmdlet - “Add-MailboxFolderPermission”.
This cmdlet is available in on-premises Exchange and in the cloud-based service. Some parameters and settings may be exclusive to one environment or the other.
Syntax:
Add-MailboxFolderPermission
[-Identity] <MailboxFolderIdParameter>
-AccessRights <MailboxFolderAccessRight[]>
-User <MailboxFolderUserIdParameter>
[-Confirm]
[-DomainController <Fqdn>]
[-SendNotificationToUser <Boolean>]
[-SharingPermissionFlags <MailboxFolderPermissionFlags>]
[-WhatIf]
[<CommonParameters>]
Examples:
Add-MailboxFolderPermission -Identity ayla@contoso.com:\Marketing -User ed@contoso.com -AccessRights Owner
# This example grants the Owner role to Ed on the Marketing folder in Ayla's mailbox.
Add-MailboxFolderPermission -Identity ayla@contoso.com:\Calendar -User julia@contoso.com -AccessRights Editor -SharingPermissionFlags Delegate
# In Exchange Online, this example adds Julia as a calendar delegate to Ayla's mailbox, but without access to private items.
Add-MailboxFolderPermission -Identity ayla@contoso.com:\Calendar -User laura@contoso.com -AccessRights Editor -SharingPermissionFlags Delegate,CanViewPrivateItems
# In Exchange Online, this example adds Laura as a calendar delegate to Ayla's mailbox
with access to private items.
For further information on mailbox folder permission, please refer to the MSDN article titled Add-MailboxFolderPermission.