There is no problem using tongue-in-cheek as an adverb.
It certainly would sound odd (and possibly leave people tongue-tied) if you attempted to add an -ly to tongue-in-cheek in an effort to build the adverb. :lol:
Although I don’t see it specifically listed as a noun in the dictionary entries I’ve checked so far, I wouldn’t rule out the possibility that tongue-in-cheek might also be used as a noun (on rare occasions).
But the most frequent usage would be as an adjective.
Have you seen many people literally put their tongue in their cheek when joking? I only know a few who (unconsciously?) do that and, curiously, they are men. Don’t you find the ensuing lopsided smile cute?