Pony Cart Duty was initially unavailable in the SE version of SLaVE. As of v9.2 it is available.
It's always been in LE and is now improved in LE as of v9.2.
Skyrim is barely able to support this feature, so don't expect perfection, particularly in SE. The tips that follow will help.
Methods to Start Cart Duty
Timed Cart Duty - Speak to Nocturnal. You can shout and then talk to her before she summons away. If outdoors, she will have a dialogue option for Cart Duty. For your submissiveness, she will remove any Breast Totrutre Collar or Asphyxiation Collar. Take her for a ride for the time required (5mins by default, can change in MCM). You can do so while travelling to another quest objective. The timer excludes periods of paused cart duty.
Alternatively, speak to Nocturnal by worshiping one of her Altars found outside entrances of Nocturnal-related locations. This is the same as #1, but easier to get her attention.
Maiden Retrieval Quest - When you rescue a maiden for Nocturnal, you give the maiden a ride home using the cart. This can be turned off in MCM. Going home means... travelling to one of the Deadric shrines where you've earned trust.
Towing the Cart - Once activated, the cart can be towed any place, other than indoors, however, it's not wise to pull it in cramped areas such as within city walls. Try if you like to find out why. Eventually, it will go physics berzerk and maybe kill you. Don't report it as a bug, this was never intended to work in Skyrim at all, lol... and there's no fixing it.
Pausing Cart Duty - Toys&Love's EndScene hotkey (End key by default ), will pause Cart Duty. Entering an interior, fast travel, or getting into combat, will automatically pause. Standing still for more than about 10 seconds will also auto-pause.
End Cart Duty Early - There's an option to end cart duty in the MCM. If you do this during Maiden Retrieval the quest will properly complete. Your Pony Gear will unequip.
The Cart will Follow You - If you fast-travel, including by carriage, the cart will reappear where you go, and you can activate it again to pull it.
You are Tethered to the Cart - When you pause the cart is still tethered to you... a very long tether, but if you move away from it too far, you will magically be pulled back to it. There's no walking away from cart duty pony girl! Giddy up!
The Yoke Removes During Combat - You can kick for defence when the yoke is on, but it will quickly be removed by Nocturnal and re-equip once not in combat, and no weapon/spell is drawn.
The Cart May Flip Upside Down - If this happens, to right it, you can pause Cart Duty, then use SLaVE MCM - Pony Girl - Move Cart To Me, then re-activate.