There isn't a transmission/gearbox, but Teslas undoubtedly have differentials in some form which is where you would definitely find oil. I change the axle oil about once a millennium in my Jeep, so Tesla owners probably don't even think about it. To your point, though, there is probably a single reduction gear to lower the rotation speed of the motor and it might need oil. (Reverse does not require reconfiguring anything in an electric system, given that one just flips polarity. Electric motors have different torque characteristics which obviate the need for traditional gearing.)

I'm guessing service on Teslas is the usual: brake pads every century given that regenerative braking does most of the work, greasing bearings every now and then, coolant, tires, wipers. Internal combustion is extremely complicated in comparison.

No you don't need a gearbox.