The answer to this question is subject to debate – so much debate, in fact, that I became sorry I ever asked the question. The answer that I originally posted was:
If we refer to words that lack "true" vowels (a, e, i, o, and u), the answer is that a number of them do; among them are: hymn, rhythm, myth, sylph, and syzygy (the alignment of three celestial objects). These words contain the so-called semivowel y and none of the true vowels.
The debate is further complicated by the view held by some people that certain uttered sounds that have distinct meaning (e.g., Shhh! Grrr! Psst! Zzzz!) are true words and therefore qualify as words without vowels (or, for that matter, semivowels). If we disqualify these words and words ending in y (together with acronyms such as RSVP and abbreviations such as P.S. and P.M. ), every English word contains at least one of the principal vowels.
After I posted the above answer, several people disagreed. Among the dissenters were those citing cwm, a Welsh word for "valley," which does have a listing in some English dictionaries. Therefore, I probably needed to clarify further to rule out words that are strictly dialect, especially Gaelic or Welsh words in which w may have a vowel sound. Someone else suggested that nth was a word with no vowels. But the n in nth stands for any number, so then we might as well argue that 1st, 2nd, 3rd, 4th, and so on are vowel-less words.
The whole matter became silly and trivial, and I am now asking people not to send me suggestions of obscure vowel-less words. Having endured several such suggestions, my final answer to the question is: Who cares?