State your interest outright, in a clear and concise manner, preferably in person, face to face.
I understand that JOAN recently retired, leaving a vacancy in JOAN'S
POSITION. I know I have only been with the company for four months,
but I believe I can take on that responsibility and do the job well.
Will there be a formal process for hiring a new JOAN'S JOB TITLE? If
so, I would greatly appreciate an opportunity to apply.
I would shy away from emails, as they are impersonal and can be ineffective, but sometimes they are the only option. If you don't work in the same building as the hiring manager, a phone call would be better than an email, but don't leave a voicemail if you don't get an answer. Wait and call back again later.
From my perspective there is nothing to be lost in asking. Best case scenario you get a promotion. Worst case scenario you get denied, but management understands that you have an interest in moving up.
You've got nothing to lose.