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 (circled), 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.  Do not remove the cover yet.

Slide the front cassette door toward the hinge and angle the black metal tabs so that they slide out of the player body.  There is a hook on the bottom edge that will catch on a tab inside the back cover.  Angling them to the orientation pictured here will allow them to clear the tab.

Once the cassette door is free, the back cover can now be removed.  Lift up the free edge a bit and slide the cover toward the play/stop/forward/rewind buttons.  Note that the gray plastic side switches will fall out when you do this.  They are interchangeable.

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.

Remove the two screws (circled) holding down the main circuit board and hinge the board up out of the way.  A piece of tape can be used to hold the board in a vertical position.  Note that the two screws are different and should not be mixed up during reassembly.

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

There is a green flat-flex cable glued across the top of the supply reel, that must be detached to allow movement of the take-up reel.  A heat gun can be used to gently warm and soften the adhesive, and the cable can be slowly and carefully pulled away from the surface.

Along the right edge of the player is another green flat-flex cable with two big capacitors mounted on it.  This also needs to be detached to allow movement of the take-up reel.  Again, a heat gun can be used to soften the adhesive so that it can be gently pulled free.

The capstan flywheel needs to be removed to allow movement of a flange that is screwed on top of a take-up reel mount.  This is done by prying off the bushing (circled) at the end of the capstan 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 4 circled screws to detach the take-up reel.  Now that the flat-flex cables have been freed, the take-up reel assembly may be slid out from underneath the reversing pulley.

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.

Once the take-up reel has been freed, it can be carefully folded back to reveal the gear (already removed in this image).  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 toward 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 "Detach Main Circuit Board" image).  The electrical contacts should probably be cleaned while you are in there.

Remove the two circled screws that hold the power supply cover in place.

Once the two screws have been removed, the power supply cover can be slid downward and removed.  The white plastic side piece will need to be gently bent back to release a tab holding the cover in place.

The black plastic DC barrel jack must be freed to remove the supply reel.  It is held in place by a friction fit, with a tab on each side sliding into a slot on the jack.  A pair of fine-tipped pliers can be used to gently rock the jack back and forth while pulling upward to release it.

Remove the two remaining screws holding the supply reel assembly in place (circled locations) and slide it out from under the reversing pulley.

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.

Once the supply reel assembly is freed, it can be partially flipped over to expose its gear.  Just like with the take-up 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.