I'm not sure what the conventions are, but there are a number of words that use 'y' in place of vowels. I believe some commentators consider y and w as de facto vowels, in which case these examples would not apply, but here are some possibilities:

pygmy

gypsy

(Both have alternative spellings, with 'i', though.)

rhythm (and its extension: rhythms) is about the longest I have come across. Also (are there enough 'y's in Scrabble for this?): syzygy.

The longest string of letters without a vowel (or a fake vowel like y or w) occurs, I believe, in the compound word: laTCHSTRing.

Hope this helps.