EXISTS clause is not availaible for MySQL . Anyways you don't need it , the AND condition in WHERE clause performs the checking part whether values exists in source table Sells
.
Try this
INSERT INTO AsksFor (Username, ItemName, ItemManufacturer)
SELECT DISTINCT 'Harish',ItemName, ItemManufacturer
FROM Sells
WHERE ItemName='zkoxtlv93' AND ItemManufacturer='tbzrt93'