There is a sentence, semantically nonsensical but syntactically valid, making use of every letter. It's shorter in length than the ubiquitous "the quick brown fox...", but I can't recall what form it takes.

You should also check out Ella Minnow Pea by Mark Dunn, a "progressively lipogrammatic epistolary fable"* about a small island whose inhabitants are prohibited from using letters of the alphabet in their writing and speech as they fall off a statue of the "author" of the Lazy Dog sentence. Their language can only be returned to normal if someone can find a 32-letter (or less) pangrammatic sentence to replace the Lazy Dog one. It's a fun read, but it's also thought provoking.

*That is, a fable in the form of letters written between the characters, in which more and more letters of the alphabet become forbidden.