What an odd question! Why on earth could you want to know this? Shakespeare could hardly avoid using a word without vowels, as we use them all the time, every day. Nearly all of them (perhaps all of them) involve replacing the vowel sound (often an 'i' - 'aye') sound with a 'y', which substitutes in the same sound as a vowel combination might.

So some words that you might look out for:

Shy gypsy slyly spryly tryst by my crypt.

Exactly! No vowels anywhere! And you can hardly avoid using "by" and "my", particularly not when so many Shakespearean oaths ("by my honour", "by my troth" and so on) involve using them both together, as you have to swear by my whatever. As in this example:

By my sword, Beatrice, thou lovest me.
(Much Ado About Nothing, 4.1)

There are more examples of this, lots more. And also lots of examples of moments at which Shakespeare will use a word without vowels. Here are two - by no means all:

Come, ho! and wake Diana with a hymn!
With sweetest touches pierce your mistress' ear,
And draw her home with music.
(Merchant of Venice, 5.1)

A shy fellow was the duke: and I believe I know the cause of his withdrawing.
(Measure for Measure, 3.2)

Hope it helps!