If the shield doesn't vibrate, how could it produce sound? This would necessitate something vibrating to move the air.
The only thing that I can think of that is plausible, is that vibranium is able to quickly dampen vibrations, and isn't just immune to them. This would then allow it to produce sounds.
There are sounds when Captain America's shield is struck due to the metallurgical construction used in the manufacture of the shield. Since the shield is not composed of pure Wakandan vibranium, the energy and vibration dampening power is lessened.
The answer lies in the materials used to make up the shield. Captain America's shield is not composed completely of vibranium:
In either case, it is the impure nature of the alloy that allows the shield to absorb vibrations and kinetic energy best at the center of the shield, making it capable of surviving blows of incredible force (like Thor's hammer) and to a lesser extent near the edges. Since there is less vibranium near the edges, the shield would make the characteristic "kang" when bouncing off of objects, redirecting the energy of the contact, and would generate short term vibrations (the sounds of ricochets) when struck by bullets.
A curious effect from the Marvel Cinema shield was when it was struck by Thor's hammer, it dispersed the energy of the blow along the edges of the shield knocking Thor backward and destroying a section of the nearby forest around Captain America. He, of course, was unharmed. This was certainly unexpected from my perspective as long-time fan of Captain America and unprecedented in terms of effects from his shield. I am curious where they will take this effect in the future.
Contrary to the accepted answer, the MCU shield is only ever stated as being made of Vibranium, not an alloy, which means that the noise we hear can't be definitively dismissed due to its "impure nature".
STEVE: What’s it made of?
HOWARD STARK: Vibranium. Stronger than steel and a third of the weight.
Steve slides the shield onto his arm.
HOWARD STARK: (CONT’D) It’s completely vibration absorbent. Should make a bullet feel like a cotton ball.
- Captain America: The First Avenger
In fact, evidence would seem to stand against this explanation, since Black Panther's Vibranium claw attacks can also be heard. Lacking evidence that the shield is anything but Vibranium, there are still other potential explanations.
Similar to the tropes of audible sharpness / audible gleam, the sound could be added in for the audience's benefit. If it were truly completely silent, it'd mess with our expectations about how materials interact, and could be a bit disconcerting. We do know Vibranium is capable of being completely sound absorbent from Black Panther's sneakers, which would suggest that the only time we're meant to not hear it is when it's directly relevant to that property. Maybe they really do fully absorb sound, but like scenes shot in space, sound is added in for our benefit.
We know Vibranium can be used to make something truly "completely vibration absorbent" as we saw with the sneakers above. But that's never stated to be a property of the Vibranium alone, and is very likely the product of Wakanda's advanced scientific understanding of the metal and its many uses. Perhaps it's natural state is one of enhanced vibration absorption, but not "complete". This meshes well with the other properties we witness, such as it's ability to bounce off targets and not simply drop to the floor on the first impact, and the fact that Stark says bullets "feel like a cotton ball" instead of feeling like nothing at all. You still feel some degree of the impact, so it's not "complete".
In the end the best explanation is probably the rule of cool in action, to allow the shield to exhibit all the awesome properties it does without the reality of the downsides such properties would impose.
