funny little elec-motor buzzing noise on and off

It's probably the brake booster. Pump the brakes a couple times while stopped/parked and that will probably trigger it.

There's also a periodic soft beeping sound that I think comes from the HVAC. Can anyone explain what that is?

 

“Funny”?
Funny “ha ha” or funny “strange“?
Does it “amuuuuse” you?

Sorry, I couldn’t help my self. Great movie.

 

The brake pedal is given a vacuum assist to make the pedal easier to push (AKA power brakes). In traditional cars, vacuum is sucked of the intake manifold. But in modern cars, an electric vacuum pump provides the assist. Thus, every time you pump the brake pedal, the vacuum pump operates.

 

It is actually a pump plus accumilator, that allows the pump to not need to run all the time. Though any engine with a throttle plate will also use that vacuum when available. Aside from full hybrids, diesels, and turbos, a car likely doesn't need this system.

 

There is no vacuum used in the Prius brake system. There is an electric pump that forces brake fluid into a high-pressure accumulator.

If you want to count the vacuum downstream of the throttle making the engine hard to spin during engine braking, then yeah, there is that vacuum.

 

Ya beat me to it!

 

Okay, I stand corrected. The point is the brake assist is making the noise...

 

The 4runner has been using a similar system since 2003 (and it probably wasn't the first). I'm guessing the electric pump allows more brake pressure to be used for when it automatically applies the brakes for traction control, rather than a vacuum brake booster that might not be able to recharge as quickly.

Tacomas from the same era used a vacuum brake booster.

 

Traction control systems mostly use the ABS pump.
There are multiple ways of operating. The electric system that Toyota uses is useful since the engine doesn't need to operate.
Diesel engines and turbocharged petrol engines often use a camshaft driven vacuum pump to provide vacuum.

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Hmm, well, we can say there is a brake assist pump, and it does make a funny little elec-motor buzzing noise on and off.

So it certainly might be what's making the OP's noise.

But when saying that's what is making the OP's noise, most of us aren't right there with the OP's car.