Vou começar dizendo que você realmente precisa normalizar seu banco de dados. Mas isso é outra questão. Enquanto isso, você pode usar esta consulta. Eu mostrei apenas para quatro ingredientes, você precisaria estendê-lo para 15 para o seu caso.
SELECT *
FROM barbot_drinks d
WHERE (ingredient_1_dispenser = '' OR
EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_1_dispenser AND i.ingredient_code = d.ingredient_1_code)) AND
(ingredient_2_dispenser = '' OR
EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_2_dispenser AND i.ingredient_code = d.ingredient_2_code)) AND
(ingredient_3_dispenser = '' OR
EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_3_dispenser AND i.ingredient_code = d.ingredient_3_code)) AND
(ingredient_4_dispenser = '' OR
EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_4_dispenser AND i.ingredient_code = d.ingredient_4_code))
Com seus dados de amostra, isso produz a seguinte saída:
id drink_id drink_name drink_image drink_desc ingredient_1_dispenser ingredient_1_code ingredient_1_dosage ingredient_2_dispenser ingredient_2_code ingredient_2_dosage ingredient_3_dispenser ingredient_3_code ingredient_3_dosage ingredient_4_dispenser ingredient_4_code ingredient_4_dosage
1 vdk_org Screw Driver screw_driver.jpg Screw Driver description... bottle vdk 1 valve oj 2000
2 vdk_cran Cape Cod cape_cod.jpg Cape Cod description... bottle vdk 1 valve cbj 2000
3 dry_mrtn Dry Martini dry_martini.jpg Dry Martini description... bottle vdk 2.5 bottle vrmth .5 valve orgbit 200
Editar
Com base em discussões um pouco extensas com o OP, a consulta foi refinada conforme abaixo. Essa consulta exige que o dispensador, o código e a dosagem estejam todos presentes para cada ingrediente da bebida. Um SQLFiddle está disponível.
SELECT *
FROM barbot_drinks d
WHERE (ingredient_1_dispenser = '' AND ingredient_1_code = '' AND ingredient_1_dosage = '' OR
ingredient_1_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_1_dispenser AND i.ingredient_code = d.ingredient_1_code)) AND
(ingredient_2_dispenser = '' AND ingredient_2_code = '' AND ingredient_2_dosage = '' OR
ingredient_2_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_2_dispenser AND i.ingredient_code = d.ingredient_2_code)) AND
(ingredient_3_dispenser = '' AND ingredient_3_code = '' AND ingredient_3_dosage = '' OR
ingredient_3_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_3_dispenser AND i.ingredient_code = d.ingredient_3_code)) AND
(ingredient_4_dispenser = '' AND ingredient_4_code = '' AND ingredient_4_dosage = '' OR
ingredient_4_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_4_dispenser AND i.ingredient_code = d.ingredient_4_code)) AND
(ingredient_5_dispenser = '' AND ingredient_5_code = '' AND ingredient_5_dosage = '' OR
ingredient_5_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_5_dispenser AND i.ingredient_code = d.ingredient_5_code)) AND
(ingredient_6_dispenser = '' AND ingredient_6_code = '' AND ingredient_6_dosage = '' OR
ingredient_6_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_6_dispenser AND i.ingredient_code = d.ingredient_6_code)) AND
(ingredient_7_dispenser = '' AND ingredient_7_code = '' AND ingredient_7_dosage = '' OR
ingredient_7_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_7_dispenser AND i.ingredient_code = d.ingredient_7_code)) AND
(ingredient_8_dispenser = '' AND ingredient_8_code = '' AND ingredient_8_dosage = '' OR
ingredient_8_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_8_dispenser AND i.ingredient_code = d.ingredient_8_code)) AND
(ingredient_9_dispenser = '' AND ingredient_9_code = '' AND ingredient_9_dosage = '' OR
ingredient_9_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_9_dispenser AND i.ingredient_code = d.ingredient_9_code)) AND
(ingredient_10_dispenser = '' AND ingredient_10_code = '' AND ingredient_10_dosage = '' OR
ingredient_10_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_10_dispenser AND i.ingredient_code = d.ingredient_10_code)) AND
(ingredient_11_dispenser = '' AND ingredient_11_code = '' AND ingredient_11_dosage = '' OR
ingredient_11_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_11_dispenser AND i.ingredient_code = d.ingredient_11_code)) AND
(ingredient_12_dispenser = '' AND ingredient_12_code = '' AND ingredient_12_dosage = '' OR
ingredient_12_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_12_dispenser AND i.ingredient_code = d.ingredient_12_code)) AND
(ingredient_13_dispenser = '' AND ingredient_13_code = '' AND ingredient_13_dosage = '' OR
ingredient_13_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_13_dispenser AND i.ingredient_code = d.ingredient_13_code)) AND
(ingredient_14_dispenser = '' AND ingredient_14_code = '' AND ingredient_14_dosage = '' OR
ingredient_14_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_14_dispenser AND i.ingredient_code = d.ingredient_14_code)) AND
(ingredient_15_dispenser = '' AND ingredient_15_code = '' AND ingredient_15_dosage = '' OR
ingredient_15_dosage != '' AND EXISTS(SELECT * FROM barbot_ingredients i WHERE i.active = 1 AND i.dispenser_type = d.ingredient_15_dispenser AND i.ingredient_code = d.ingredient_15_code))
