Possibly. You can send information through power lines for example. There are consumer devices that use it for like local internet connections, and I think power companies have started using something like it to replace meter readers.

There are multiple ways to transmit radio frequencies through computer monitors (e.g. this), and communication via ultrasonic sound which we can't hear.

See a recent MIRI paper.

A narrow AI, "tasked with designing an oscillating circuit, re-purposed the circuit tracks on its motherboard to use as a radio which amplified oscillating signals from nearby computers."

One of those things we can't know for sure, because there could hypothetically exist a new physical law we don't know yet. The AI could somehow learn about this law -- even if it cannot do experiments, it could somehow derive it from the first principles... ahem, Solomonoff priors.

But I guess that is rather unlikely.

Then, there is the possibility of using the familiar laws of physics and the existing hardware (which is what you asked). Seems to me that this kind of output would be too noisy. I can imagine the AI using the fans to create some resonation and maybe literally break the box... but since the AI is at that moment not a physical entity but only a pattern existing in the computer memory, that would be equivalent to suicide.

Okay, with a really insane computing power the AI could hypothetically measure the positions of particles in the air, and use the fans to manipulate them... now this would depend on whether the AI can gather the information about the particles in its environment faster than the information is lost because of e.g. ventilation in the room which keeps bringing new particles with unpredictable positions and speeds. At some moment the AI would be limited by the fact that it does not have literally infinite computing power.

(AI with literally infinite computing power and infinite computing speed could probably do almost anything. It could use Solomonoff priors to model all possible universes, use all its data to select the most likely one, and for any possible action X it could make, it could calculate whether its desired goal is more likely to happen if it does X or if it does non-X. Thus the probability of the goal would keep increasing; the question is how fast, and whether that growth would have a limit lower than 100%. Maybe the AI would require millenia to strategically modify the particles in the air to bring the desired outcome through e.g. a series of almost invisible social changes; but maybe it will be disassembled sooner so it cannot complete its plans. But literally infinite computing power and speed is physically impossible... according to our knowledge of the laws of physics.)