Is there a way to contact the local office to speed up the processing?
No. The only portal where you can request urgent processing is the webform.
Can urgent processing be justified with applying for (a) federal job(s) without a job offer?
I don't think so. You'll need a offer that says the offer is confirmed if citizenship is obtained in most cases. There might be edge cases here and there but at a minimum, IMO, you need to be somewhere in the process (shortlisted, for example). However, there is no harm in giving it a try. Try applying for urgent processing and see if it works.
There are federal jobs I would be qualified for but they need Canadian citizenship to be able to apply for.
Do you guys have any idea whether this might be a legitimate reason for urgent processing? Has anybody had any experience with urgent processing? Your help is truly appreciated.
Give this a read.