First you need to make a ticket field for the requester to enter when the submit the ticket. Then you’ll need to create a new Target for Twilio (you can’t use the one that ZenDesk offers you as that just lets one person receive a text). The target will need to use the API info from Twilio:

URL: https://api.twilio.com/2010-04-01/Accounts/**YOUR TWILIO ACCOUNT SID GOES HERE**/SMS/Messages?From=**YOUR TWILIO PHONE NUMBER GOES HERE**&To=**YOUR TICKET FIELD GOES HERE**

Your ticket field will look something like this: {{ticket.ticket_field_123456}}

Your URL should like something like this:
https://api.twilio.com/2010-04-01/Accounts/AbCdEfGhIjKlMnOp12345/SMS/Messages?From=+12125551234&To={{ticket.ticket_field_123456}}

Method: Post
Value: Body

Basic Authentication:
Username: (Your Twilio Account SID)
Password: (Your Twilio Auth Token)

Finally, to send messages to SMS, you’ll need to create Triggers. You’ll set your triggers as you’d like. The “Perform these actions” should include “Notify target” and the name of your Twilio target you created above. Whatever you type in “Message” will be sent by SMS.