Is the sentence "You rise groggily, the campsite a casualty of merriment." grammatically correct?
Yes, it is correct. The campsite a casualty of merriment is an absolute phrase.
An absolute phrase doesn't modify anything in the sentence but rather sets a scene. It consists of a noun/pronoun and a participle phrase:
I slept fitfully, the reindeer prancing on the roof.
If the participle is a form of to be, it can be omitted:
Reindeer [being] on the roof, I slept fitfully.
That's like your sentence:
You rise groggily, the campsite [being] a casualty of merriment.
(In case you were wondering about you rise groggily — it uses the narrative present.)