Here is a list of woodwind instruments. Do you know of any (non-open) reed instrument that produces a note without anyone blowing air through them? Imagine a clarinet being played on someone's lap pouring out a melody. That would be very, very remarkable indeed.
Our ability to produce sound from our throats is in theory like a reed instrument in music. Let's take the flute.
The flutist blows a rapid jet of air across the embouchure hole. In the flute, the air jet produces an oscillating (vibrating) component to the air flow. Once the air is vibrating, the energy is radiated as sound out of the end and any open holes. The column of air in the flute vibrates much more easily at some frequencies than at others (i.e. it resonates at certain frequencies). These resonances largely determine the playing frequency and thus the pitch, and the player in effect chooses the desired set of resonances by choosing a suitable combination of keys. (More on that here.)
Our vocal chords don't vibrate without the movement of air through them. They just tense and relax to different degrees. There is no sound made if you go through all the motions of saying yes but don't move any air. The moment you move even a small amount of air, you'll get a sound. Furthermore, vibration of the vocal chords is not enough for speech. The noise must go through a chamber where a particular noise will be made to change to resemble speech. Say ahhhhhh and touch the tip of your tongue to the roof of your mouth. The sound is changed significantly. Now, add your cheeks, lips, teeth, and pharynx all changing shapes and you have language.
Our voices are a wind instrument (not like an electronic speaker. For that, look to the electrolarynx). The air column moving across the vocal chords is the air jet. The vocal chords are what sets the air to vibrate. Our "mouth" (all of those I mentioned above" affect the vibrations to produce the desired sound. That's speech.