You can open the cassette compartment and visually confirm that one or both of the supply reel (left) or take-up reel (right) has a broken gear.  You should be able to see a crack in the gear, and usually a large gap that has opened up.

Unscrew the center cover of the volume dial by placing something pointy into one or both of its indentations to turn it.  Remove the dial, making sure to not lose the clear plastic washer that is underneath it.

Remove the 4 screws (circled) on the bottom edge of the player and fold open the bottom cover.  Note the "longer" screw heads at the corners, for when you replace them.

Remove the now-exposed 4 screws (circled) on the back cover and remove it by lifting up the free edge a bit and sliding the cover toward the play/stop/forward/rewind buttons.  Note that the gray plastic side switches will fall out when you do this.  There are two different designs, and they can all fit back in place in only one order.

When reassembling the player, gravity is your friend.  Orient the player so that the side switches are pointing down, lay the switches inside the back cover, and close up the player while keeping everything aligned.  It's fiddly, but do-able.

Unsolder the tan and black wires (circled) near the Tape/Radio switch at the bottom left of the board, with the player oriented with the Play/Stop buttons toward you.  Gently peel the wires back from the adhesive holding them in place and completely free them from the main circuit board.

While you're in here, it is a good time to clean out the main drive belt (arrow).  It will almost certainly have turned into a gooey mess if it is still the original.

Remove the two screws holding down the main circuit board (circled, left) and the two screws in the power supply channel (circled, right) to free the main circuit board.

Gently lift up the front edge of the circuit board, sliding both battery contacts up and out of their holders, and pulling the connecting wires free from their adhesive.  A piece of tape can be used to hold the board up in a vertical position and out of the way.

Remove 4 screws from the supply reel/motor assembly on the right (from circled locations).

Note that there is some movement of the supply reel around these screws, so care should be taken during reassembly to center it (when viewed from the other side) to avoid rubbing during operation.

Now that the supply reel is detached, it can be folded out to the right.  If this is the gear being replaced, you can pry/peel the old gear off of the center metal reel and snap on a new gear.  In this orientation, the exposed gear teeth should be on the bottom and the thin, toothless edge should be on the top facing you.

If this is the only broken gear, then you may reverse the above steps to reassemble the Walkman.  When replacing the main circuit board, make sure the forward/reverse leaf springs fit correctly between the plastic levers (pictured in the "Unsolder Wires" image).  The electrical contacts should probably be cleaned while you are in there.

If you still need to replace the take-up gear on the left, then you must disassemble more parts of the player that block access.  You can either remove the entire tape transport assembly or a small subset that is just enough to free the take-up reel (with a bit of difficulty).  We'll take the second approach here.

Begin by removing the capstan and its flywheel.  You can free the capstan by prying off the bushing (circled) at the end of the shaft.  This bushing can be easily broken or cosmetically damaged, so a replacement is included with each gear.  Once the bushing is removed, the capstan and its flywheel can be slid out.  Do not lose the small plastic spacer that rides on the capstan shaft against the flywheel.

Remove the two remaining screws (circled) on the left edge of the take-up reel assembly.  The other two screws overlapped the supply reel/motor and have already been removed.

Note that there is some movement of the take-up reel around these screws, so care should be taken during reassembly to center it (when viewed from the other side) to avoid rubbing during operation.

By carefully bending the green flat-flex connections, you can push the take-up reel assembly back and to the left to angle it out from underneath the reversing pulley.  This is difficult and takes some work to avoid breaking or tearing anything, but it can be done.

Once the take-up reel assembly is freed, it can be flipped over to expose its gear.  Just like with the supply reel, this gear can be pried/peeled off the center metal reel.  The replacement gear can simply be snapped in place.  In this orientation, the exposed gear teeth should be on the bottom and the thin, toothless edge should be on the top facing toward you.

From here, all of the above steps may be reversed to reassemble your Walkman cassette tape player.