Error after entering the New-MailboxExportRequest parameters:

Couldn’t connect to the source mailbox.
+ CategoryInfo : NotSpecified: (0:Int32) [New-MailboxExportRequest], RemotePermanentException
+ FullyQualifiedErrorId : 4E7EDE9C,Microsoft.Exchange.Management.RecipientTasks.NewMailboxExportRequest

Exchange 2010 relies of MAPI when performiny any mailbox export. I had to enable MAPI.

MAPI