Vibrato is a musical technique that pulses or 'vibrates' a tone while playing, producing a kind of "wobbling" sound that is pleasant to hear. There are two steps required to use proper vibrato - good posture and deep (diaphramic) breathing.
A simple exercise to check your posture is to stand with your upper back, heels of your feet, and the back of your head against the wall. Make sure that you are standing tall with your shoulders back. There should be a small gap between the wall and your lower back, which you can check by placing your hand in that gap.
Diaphramic breathing can be slightly difficult at first, but it is helpful to visualize breathing from deep in your belly. When you take a deep breath, your belly should rise while your upper body and shoulders remain stationary. Practicing breathing this way several times a day.
To develop your vibrato, prepare to play a single note on your ocarina (preferably somewhere in the mid-range like F or G). Then take a deep breath, and blow, holding that single note without blowing too strongly. The longer you can hold that note, the easier it will be for the vibrato to "flow".
Keep in mind: the vibrato will not start immediately. It will take at least 1-3 seconds for the vibrato to start naturally. You should not at any point feel like you are forcing your throat or mouth to pulse or vibrate. It is all in the breath and diaphragm! It also may not occur for the first few times that you attempt it, but if you practice your posture and deep breathing, it will come with time.