Count Data on Third Table Through Link
Here's what I came up with (SQLFiddle):

SELECT
  st.SupportTicket_ID,
  st.Submitting_User,
  st.Submitting_Department,
  SUM(
    CASE
      WHEN d.Type = 'Cellphone' THEN 1
      ELSE 0
    END
  ) AS [Mobile Devices],
  SUM(
    CASE
      WHEN d.Type = 'Computer' OR d.Type = 'Disc' THEN 1
      ELSE 0
    END
  ) AS [Computers],
  SUM(
    CASE
      WHEN d.Type = 'Camera' OR d.Type = 'Video Cassette' THEN 1
      ELSE 0
    END
  ) AS [Audiovisual]
FROM SupportTicket st
JOIN Link l 
  ON st.SupportTicket_ID IN (l.Entity_ID1, l.Entity_ID2)
JOIN Device d 
  ON d.Device_ID IN (l.Entity_ID1, l.Entity_ID2)
WHERE 
  l.Link_ID LIKE 'CAS%'
  AND st.Type = 'Tech Support'
  AND st.Date_Submitted &gt;= CONVERT(datetime, '20140201', 112)
  AND st.Date_Submitted &lt; CONVERT(datetime, CONVERT(date, CURRENT_TIMESTAMP))
GROUP BY
  st.SupportTicket_ID,
  st.Submitting_User,
  st.Submitting_Department
ORDER BY
  st.SupportTicket_ID;


The basic idea is to compute the sum of an expression that returns 1 if the current row has the required type, and zero otherwise. It's probably possible using PIVOT syntax as well.