I heard you're not allowed to walk around while eating something.

If you're in places like Kamakura’s Komachi Dori or a place that asks you not to do it, just don't do it. It's not seen as good manners normally so it's best to avoid in general. But there are certain areas that allow or encourage 食べ歩き tabearuki (eating while walking). Just see what everyone else is doing or pay attention to signs or look up places that allow it. For example, Yanaka in Tokyo! :)