Ƭᴇcʜιᴇ007 is unfortunately correct: there's no way to set up a rule that will permanently delete an email while Outlook is not running. Your options are either a) create a client-side rule in Outlook that will process all the permanent deletions at once when you next open it, on your return (prior to which, your inbox will fill up with crud), or b) create a server-side rule, via Outlook Web Access, that will delete those emails so that, when you next open Outlook, they will be in your 'Deleted Items' folder. The latter is likely preferable here.
Note: I realize you said "there is absolutely no opportunity to use a different client at all ever", but it's unclear whether that includes the web-based version of Outlook Web Access (since, after all, that's still Outlook). OWA comes with Microsoft Exchange and while I'm sure it's possible that there are organizations out there that use Exchange but do not implement OWA, I've yet to come across one.
1Just out of curiosity: Why is it so important to permanently delete the mail right away? Why not just delete them when you're back? Or just have your mailbox deactivated while you're away? Also, won't you need some of the mails when you are back? – sleske – 2015-09-06T22:52:03.033
1Because my inbox has a size limit and when the hard limit is reached I can no longer receive mail. – Lisa – 2019-02-13T01:29:02.703
1Thanks for the reply, and nice that you're still around :-). From what you write, that frankly looks more like a workplace problem. If the hard limit makes you lose mail, that's something your sysadmin or boss should address. Either the mail really is important, then you need more space - or it is spam or stupid newsletters, then they should help you to filter/delete incoming mail (maybe install a server-side spam filter, or whatever). Just tellling you to read your email, but not giving you enough space is not gonna work. – sleske – 2019-02-13T05:56:14.147