From what I'm told by a friend who used to work in a cinema here in the UK, it would depend. Some films would actually ask for their marketing material to be sent back after the film's theatrical run, so they wouldn't be able to keep any of it. In most cases though, staff would get the option to take anything they wanted and the rest would go to recycling. So your best bet would be to talk to someone in a local theatre and ask them if they think it would be possible, and if you could leave your contact info so they can get in touch if and when they have anything for you. I'd definitely say to do this before the film stops being shown, because the longer you wait the greater the chance that the posters will have been disposed of.
Alternatively, Allposters is a website that sells copies of a lot of film posters so if you can't get one direct you would definitely be able to pick up a copy. I know it's not what you asked but I'm just putting it out there
Good luck!