If Umby injured itself, what would it to do try and heal?
*Switches Rant mode on*
Difficult question, but I think that Umbreon is suprising tough for its size and it would take one nasty blow to severely injure an Umbreon. For servere injuries, it would prequire the aid of none other than Nurse Joy. However i believe its possible that Umbreon would regenerate slightly faster while under the presence of moonlight. Of course it can use the move Moonlight to restore its health. So an umbreon who's injured lightly and doesn't want to be fatigued by using Moonlight, might choose to bask in the moon's light, similar to how a reptile would do and let the moon do the work. I'd guess if it is a flesh wound that most likely it has cleaned off the wound beforehand (Probably via licking the affected area) and assuming that a) Umbreon is immune to its own poison sweat. and b)Umbreon is similar to a dog, fox or cat, its saliava would contain some healing enzymes, just as cat and dogs saliava has.
*Switches Rant mode off*
Or i'll come along and nurse Umbreon back to health :D
All this talk about injury reminds me of a disease I made in one or my fics that affected humans that could be caught from Umbreon. I called it Umbric Nightshade :)