You need to give full mailbox access on the mailbox explicitly to the user to access via OWA.

Example: Below command gives full access to the mailbox named TestA to the user named TestB, so user TestB can access TestA mailbox via Outlook Web Access.

Add-MailboxPermission -identity TestA -User TestB -AccessRights FullAccess

Reference: How to Enable Explicit Logons in Outlook Web Access
http://technet.microsoft.com/en-us/library/aa998830(EXCHG.80).aspx

You can run below command to give Full access to user TestB on all the mailboxes of a server “ESS-Exch702” hence user TestB can access all the mailboxes via OWA.

Get-Mailbox -Server “ESS-Exch702″ | Add-MailboxPermission -User TestB -AccessRights FullAccess

But this command gives permission to all existing mailboxes of server “ESS-Exch702” and it doesn’t apply on the new mailboxes and you need to run it again for the mailboxes which are created after running this.

Access permission which is given by this procedure doesn’t work while opening mailbox in OWA.

However, this isn’t actually meant to work. After months of working on a PS case with Microsoft, they actually admitted that this is a flaw with Outlook (i.e. a bug) and will be fixed in future releases.

Basically there is a inherited Deny permission for all users, stopping you from “blanket” full-access permission setting. You can see this if you use OWA, and try to access another mailbox after running this command.

So sadly, it will stop working sooner rather than later!

Damn you MS!