JOIN correctly no matter which end is linked
An OR appears to work fine in your fiddle example.

SELECT
  st.Submitting_User,
  COUNT(l.Link_ID) AS [Count]
FROM 
  SupportTicket st
  JOIN Link l
    ON st.SupportTicket_ID = l.Entity_ID1
    OR st.SupportTicket_ID = l.Entity_ID2
WHERE 
  l.Link_ID LIKE 'CAS%'
  AND st.Type = 'Tech Support'
  AND st.Date_Submitted BETWEEN '20140201' AND CURRENT_TIMESTAMP
GROUP BY
  st.Submitting_User, st.Submitting_Department
ORDER BY
  st.Submitting_User