The longest common English word containing none of the vowel letters a, e, i, o or u is “rhythms.”
“Rhythms” has seven letters, and its vowel-like sounds are represented by the letter y rather than by the five conventional vowel letters named in the question. In English spelling, y can represent a vowel sound, as it does in “rhythms,” but it is not included in the question’s definition of a vowel letter.
The word is the plural of “rhythm,” meaning a pattern of beats, movement or recurring emphasis. It is often cited in wordplay and spelling trivia because it demonstrates how English can write syllables without using a, e, i, o or u. “Myths,” “crypt” and “shyly” also qualify, but each is shorter.
The word “common” matters. Extremely rare terms, sound effects and abbreviations can produce competing claims; for example, “tsktsks” is sometimes listed as a longer lexical form without those five letters. It is not generally considered as common as “rhythms,” so “rhythms” remains the standard answer to this familiar trivia question.