Does the Inserted table contain Identity values?
The error is caused by attempting to insert AuditTerms when there is no corresponding primary key in TermKeys. After creating the table schema and trigger in tempdb and inserting a few records in TermKeys, I was able to reproduce it by issuing a DELETE twice on the same TermID:

DELETE FROM TermKeys
WHERE TermID = 1

DELETE FROM TermKeys
WHERE TermID = 1


gives me:


  Msg 515, Level 16, State 2, Procedure TRG_TermKeys_AuditTerms_IUD,
  Line 15 Cannot insert the value NULL into column 'TermID', table
  'tempdb.dbo.AuditTerms'; column does not allow nulls. INSERT fails.
  The statement has been terminated.


Something in the application code may be running the query twice; I can't see any other way to get a null value. The best way to avoid the error is to add a precautionary check of @@ROWCOUNT at the beginning of the trigger, since it will return the total number of rows affected by the DML statement that fired it.

ALTER TRIGGER [dbo].[TRG_TermKeys_AuditTerms_IUD] ON [dbo].[TermKeys] 
AFTER INSERT, UPDATE, DELETE NOT FOR REPLICATION AS
BEGIN
  -- If the trigger was fired but no rows were affected, do nothing 
  IF @@ROWCOUNT = 0
    RETURN;
  SET NOCOUNT ON;
  DECLARE @termid int;
  IF EXISTS(SELECT * FROM Deleted) SELECT @termid = TermID FROM Deleted;
  IF EXISTS(SELECT * FROM Inserted) SELECT @termid = TermID FROM Inserted;
  IF EXISTS(SELECT TermID FROM AuditTerms WHERE (TermID = @termid))
  BEGIN
    UPDATE AuditTerms SET UpdatedAt = getdate() WHERE (TermID = @termid);
  END
  ELSE
  BEGIN
    INSERT INTO AuditTerms (TermID, UpdatedAt) VALUES (@termid, getdate());
  END;
END;