My natural gas heating unit runs almost silent normally, but it started sounding similar last year when the draft inducer motor was failing. The draft inducer is an exhaust fan that starts up and makes sure it can pull the combustion byproducts out the vent stack before the gas starts up in a heating cycle.
I was able to identify it was this motor by pulling off the front panel and checking where the vibration came from when this noise was active. Pushing on the motor made most of the noise go away, but the mounting bolts were already tight and so was the seal. The draft inducer is a large spiral shape with a motor in the middle.
These have labels on them and also model numbers molded into the plastic. You should be able to look it up and replace it with the same or compatible unit. Typically the hardest part is wiring the motor but they use wiring harnesses that clip together, just route the wires properly.
And it should go without saying, but make sure the breakers for the heater are shut off before you start, because if the thermostat calls for heat while you're working, "you're gonna have a bad time."