Yes. Camera phones, home video cameras, DSLRs, GoPros. As long as it fits into the confidential locker (something like 4ft tall, 1.25ft wide, 3ft deep) it'll be fine.
I don't know that I'd recommend a camera for plebe summer. I don't think they can have one until PS is over. But perhaps some USNA Mids might be able to offer a little guidance.