While you are viewing a ticket you will see a 3rd tab called "forward". Simply click on the forward tab and then you will see a field appear which will allow you to type in the e-mail address of the 3rd party that you wish to send the ticket to. Now type in your message as usual. You can also assign and change the status of the ticket.
When you send out the ticket your vendor will receive the ticket along with all of the previous threads (including notes). When the vendor replies to the e-mail your customer WILL NOT be notified and the message will appear in your Teamwork Desk account as a note (and notes are never visible to the customer).
Note on attachments - Due to restrictions on the total size of e-mails (25MB) attachments from previous messages in the ticket will be added as links in to the body of the message. These files will be available download by anyone with the link and expire after 30 days.