Return Only Rows where count >= 1
Live by the cross apply, die by the cross apply

 CREATE TABLE #TableTest
 (
   dealerID INT,
   itemname VARCHAR(100),
   itemonhand INT,
   itemname2 VARCHAR(100),
   itemonhand2 INT,
   itemname3 VARCHAR(100),
   itemonhand3 INT
 );
 INSERT INTO #TableTest
        ( dealerID, itemname, itemonhand, itemname2, itemonhand2, itemname3, itemonhand3 )
 VALUES ( 1, 'ball', 0, 'hat', 1, 'sock', 0 )
  ,     ( 2, 'ball', 0, 'hat', 0, 'sock', 1 );


 SELECT tt.dealerID,
        ca.*
 FROM   #TableTest AS tt
 CROSS APPLY ( SELECT   *
               FROM     ( VALUES ( itemname, itemonhand), ( itemname2, itemonhand2), ( itemname3, itemonhand3) ) x ( item_name, item_count ) ) ca
 WHERE  ca.item_count = 1;