The rear wiper stopped working in my sister's 2010 Elantra Touring GLS back in March or so.
After confirming that switch and control module were OK (pulled the rear lid trim, removed the electrical connector from the motor, and measured correct voltage across terminals 1 and 4 when the rear wiper switch was activated), I ordered a new replacement motor assembly (PN 98710-2L100) on-line for about $115.

Replacement of the rear wiper motor was not too complicated:

1. With the rear lid closed, remove the the plastic cap (PN 98812-2L000) covering the end of the rear wiper arm, held on by snaps (Figure 1).
2. Remove the 13 mm nut (PN 13270-08007K) holding the wiper arm to the motor spline.
3. Remove the wiper arm (PN 98811-2L000) from the motor spline -- the wiper arm is a plastic part, so metal corrosion here shouldn't be a problem.
4. Remove the dust cap (PN 98712-3J000) covering the 24 mm flat nut (Figure 2).
5. Remove the 24 mm flat nut (98726-29600).
6. Remove the plastic cap (98714-1H000) covering the rubber mounting pad.
7. Remove the rubber mounting pad (PN 98713-1H000) from the motor spline.
8. Open the rear lid and remove the lower inner trim panel (Figure 3). Note that the service manual diagram for the liftgate lower trim panel is *not* correct; the panel is held in place with two screws near the top of panel where it mates with with upper trim pieces (around the window glass) and 10-12 body trim fasteners/clips; these just pull off with moderate force applied to the panel using a plastic body trim tool.
9. With the lower trim panel removed, remove the electrical connector from the wiper motor assembly.
10. Remove the two 10 mm bolts (Figure 4).
11. Remove the wiper motor assembly from the rear lid window.
12. Transfer the blue support piece (PN 98711-07000) and the two rubber mounting assemblies (PN 98716-07000) from the old motor assembly to the new motor assembly. Take care to note the orientation of the blue support piece.
13. Mount the new motor to the rear lid with the two 10 mm bolts, finger tight only for now.
14. Attach the electrical connector to the motor assembly and close the rear lid fully.
15. Operate the rear wiper for at least one full cycle to return the spindle to its correct rest position.
16. Assemble the rubber mounting pad (notch facing down), plastic cap, and 24 mm flat nut to the threaded motor assembly. Tighten the flat nut to torque spec.
17. Open the rear lid and tighten the two 10 mm bolts to torque spec.
18. Attach the rear lid lower trim.
19. Close the rear lid fully and assemble the wiper arm to the motor spindle.
20. Attach the 13 mm nut and tighten to torque spec.
21. Attach the wiper arm end cap. Job done.

Figure 1:

Figure 2:

Figure 3 (rear lid/tailgate trim):


Figure 4 (motor assembly):


Based on the appearance of the old motor assembly, my guess is that moisture entered the spindle shaft opening over time and corrosion seized the spindle shaft to the assembly housing.

Opening the old motor housing to see if this repairable will be a rainy day project someday soon.