A beakful is an amount of something which will fit into a bird's beak.
The beak is the hard part around the mouth of a bird that they use to eat and groom themselves or fight.
No, ostrich do not have teeth. There is a photoshopped picture on the web that shows a closeup of an ostrich with a beakful of teeth, but it is not a real photo. The teeth appear to be those of a canine. So if it has a beak there no teeth.