snack on (something)
To eat small amounts of something as a snack. I love watching the birds and squirrels come up to the feeder throughout the day to snack on seeds. I've been snacking on that roast ham all day, so I'm not terribly hungry.
snack off (of) something
to eat food, bit by bit, in little snacks. (Of is usually retained before pronouns.) Please don't snack off the turkey so we can get another meal out of it. Who has been snacking off of last night's roast beef?
snack on
v. To eat something as a snack: One way to lose weight is to snack on carrots instead of junk food.