Yeah, not sure about raised beds, would have to ask the apartment manager, but having never seen any in the complex in 14 years here, I assume they're not allowed (would also have to bring in extra dirt, probably the dealbreaker, because someone would have to remove it when I left).
I might try some herbs, as I'm really not much of a vegetable person, though in the past in growings herbs, certain ones (the mint family) have tended to take over...