The longest common word in Modern English is rhythms. However, the “y” does provide a vowel sound but none of the five vowels appear in the word.
The longest word which is still considered by some to be Modern English but is quite rarely used, is symphysy. It refers to the fusion of two bodies, The 17th-Century word is nearly obsolete. Again, the “y” provides a vowel sound but none of the five vowels appear in the word.
I’m not including any archaic words from Old English or Middle English. There were many long words that we now consider to be archaic that did not have vowels. They are simply not part of Modern English and I don’t think they belong on such a list.