We are taught that the vowels are "A, E, I, O, U, and sometimes Y." Very rarely, W can be a vowel--usually in words borrowed from the Welsh language. So there are at least two English words that do not contain A, E, I, O, U or Y. However, they do contain vowels! They are VERY VERY rare words. They are useful in word games like Scrabble. The two words are "cwm" and "crwth."

If there aren't any vowels, it can't be a word. I can think of some words without an a, e, i, o or u; but they have at least one y that functions as a vowel: hymn myth myrrh rhythm

There are quite a lot that have Y as the only vowel. The first one that came to mind was "syzygy!" A web search turned up and article listing these: by cry crypt cyst dry fly fry gym hymn lynch my myth nymph ply pry psych pygmy rhythm shy shyly sky sly slyly spry spy sync try tryst why wry

I see that at least one dictionary does include some "words" that express noises. I put "words" in quotation marks because I'm not sure everyone would agree that they are really words. I found "psst," and "tsk" in the American Heritage and Merriam-Webster dictionarys, and "pfft" in American Heritage but not Merriam-Webster. And I found that all of them are legal "words" when playing the Scrabble word game.

Tangentially, "strength" is common eight-letter English word with only a single vowel.