Pyelonephritis: This is a common condition that fits the symptoms provided. Pyelonephritis is an infection of the kidney, which can cause fever, vomiting, and flank or loin pain due to the location of the kidneys in the lower back.

Nephrolithiasis (Kidney Stones): While not always associated with fever, the severe pain (renal colic) and vomiting can be part of the presentation, especially if there's an associated infection.
Appendicitis: Although the pain typically starts near the belly button and then moves to the lower right abdomen, some cases can present with pain in the flank or loin area, especially if the appendix is retrocecal.
Gastroenteritis: This condition can cause vomiting and fever. If the inflammation affects the intestines near the kidneys, it might also cause loin pain.

Sepsis: Any infection that causes fever and vomiting can potentially lead to sepsis, a life-threatening condition that requires immediate medical attention.
Perforated Viscus: A hole in the stomach or intestine can cause severe pain, fever, and vomiting. Although the pain is usually more generalized, it can start in the loin area if the perforation irritates nearby structures.
Ectopic Pregnancy: In women of childbearing age, an ectopic pregnancy can cause severe abdominal pain, fever, and vomiting. It's a medical emergency.

Pneumonia: Certain types of pneumonia, especially those affecting the lower lobes of the lungs, can cause referred pain to the abdomen or loin area, along with fever and vomiting.
Pancreatitis: Inflammation of the pancreas can cause severe abdominal pain that radiates to the back, fever, and vomiting.
Retroperitoneal Hemorrhage or Abscess: Bleeding or an abscess in the space behind the peritoneum can cause flank pain, fever, and vomiting, although these conditions are less common.