Are there any English words that have no vowels?

The answer to this depends what you mean by “vowel” and “word.” There are two things we mean by the word “vowel”: a speech sound made with the vocal tract open or a letter of the alphabet standing for a spoken vowel. “Cwm” and “crwth” do not contain the letters a, e, i, o, u, or y, the usual vowels (that is, the usual symbols that stand for vowel sounds) in English. But in those words the letter w simply serves instead, standing for the same sound that oo stands for in the words “boom” and “booth.” “Dr.,” “nth” (as in “to the nth degree”), and “TV” also do not contain any vowel symbols, but they, like “cwm” and “crwth,” do contain vowel sounds. “Shh,” “psst,” and “mm-hmm” do not have vowels, either vowel symbols or vowel sounds. There is some controversy whether they are in fact “words,” however. But if a word is “the smallest unit of grammar that can stand alone as a complete utterance, separated by spaces in written language and potentially by pauses in speech,” then those do qualify. “Psst,” though, is the only one that appears in the Oxford English Dictionary.