Quick aside the person who made this dungeon and decided that it was a good idea to not allow parties to rest needs to be punched in the face multiple times. On to the question.

Is it even possible to rest in the Vale of Merdelain?

I think you CAN rest in the Vale, but it's just that the odds of your rest getting interrupted by monsters (i.e. shadows) is nearly 100%. Fear not though. Prior to the final battle, there is a room where you can rest up and renew spells before the final confrontation.