Just take the tumbler out.

Put the key in and turn to RUN (DO NOT START)

Underneath the tumbler will be a hole where you place an awl (or similar) to push the detent in and then you pull the whole tumbler out. You may have to remove the steering column trim piece to get to the detent hole.

Once removed you can rotate the key and remove it. Put it in and out and you will see the piece of metal that pops up when the key is insert. This piece of metal grounds to the inside where the tumbler goes and activates the buzzer.

This piece could be dirty. Clean with electrical contact cleaner and lubricate with some dialectic grease.

Or you could just spray the whole thing down with WD-40, let it drip dry, wipe off and reinsert. Make sure you turn it back to RUN prior to reinstalling it. And wipe down the inside where the tumbler goes before you put it back in.