My kitten's back legs are paralyzed. Can it heal without a vet?

Poor cat! If there is no deep pain present in the back legs then chances are kidding you won't regain function unfortunately. We use the pain is an indicator whether the pet will walk again. However if deep pain is present (kitty moves twitches or flinches when you pinch the back toes), then there is a chance that Katie can regain function of the limbs. However you won't know the extent of the injury without an x-ray. And without an x-ray he won't really know what treatment kitty needs. I unfortunately don't see a good solution here without proper diagnostics. Kittie may be a lost cause at this point without proper medical care for such a traumatic injury like this which undoubtedly involves the spinal cord and/or pelvis. I would highly consider Humane euthanasia for this kittie because the outcome looks bleak.