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So I am looking to interview for 3L OCI. I am currently interning at a big law firm and want to use a work product for 3L OCI as a writing sample. How can I ask for permission without affecting my chances of receiving a return offer?

I've never asked permission and it's never been an issue. Change every recognizable attribute in the writing sample. For example, I once used a motion that I wrote as a summer associate in a patent litigation case. But the memo concerned an evidence issue, and the particulars of the technology were completely irrelevant. So in addition to changing the names of the parties, companies, people, and even the courts, I changed the technology at issue completely, from say, a hard drive to an optical system for a camera phone or something like that. Literally none of the changes affected the substance of the motion. It was a "find and replace" exercise, with an occasional grammar fix here and there.

Obviously that may be a lot harder to do with work product that goes to the heart of a substantive issue.