Hyponatremic seizure due to severe dehydration: This is a likely diagnosis given the history of vomiting for several days, which can lead to severe dehydration and electrolyte imbalances, including hyponatremia. Seizures and respiratory depression can occur due to the severe electrolyte imbalance affecting brain function.

Infection (meningitis or sepsis): Infections can cause vomiting, seizures, and changes in breathing patterns in infants. The combination of symptoms could indicate a serious infection affecting the central nervous system.
Metabolic disorders: Certain metabolic disorders can present with vomiting, seizures, and respiratory changes, especially if the disorder leads to an accumulation of toxic substances that affect the brain.
Pyridoxine-dependent epilepsy: This is a rare cause of epilepsy that can be triggered by metabolic stress, such as that caused by prolonged vomiting, and can present with seizures.

Reye's syndrome: Although rare, Reye's syndrome is a potentially fatal condition that can occur in children, especially after a viral infection, and is associated with the use of aspirin. It presents with vomiting, seizures, and respiratory depression due to liver failure and cerebral edema.
Intussusception: This condition involves the telescoping of a portion of the intestine into another, which can cause severe abdominal pain, vomiting, and potentially lead to seizures and respiratory distress if not promptly treated.
Intracranial hemorrhage or stroke: These are emergencies that can present with sudden onset of seizures and changes in breathing, and must be considered, especially if there's a history of trauma.

Congenital adrenal hyperplasia: This group of inherited disorders can lead to adrenal insufficiency, which may present with vomiting, seizures, and respiratory distress, particularly during times of stress like infection or dehydration.
Mitochondrial disorders: These are a group of disorders that affect the mitochondria, often presenting with a variety of symptoms including seizures, developmental delays, and sometimes acute episodes of illness that could include vomiting and respiratory changes.
Organic acidemias: These are disorders of amino acid metabolism that can lead to the accumulation of toxic substances, causing symptoms such as vomiting, seizures, and changes in respiratory rate, especially during episodes of illness or fasting.