This vignette is a textbook description of $infantile$ $scabies$. The three pillars pointing to the answer are distribution, itch and contact history.
Distribution: unlike adults, infants with scabies show lesions on the $palms$, $soles$, $umbilicus$, axillae and even the face and scalp. The presence of vesicles on the palms and soles is almost a giveaway, because these acral sites are involved far more often in babies than in older patients.
Symptom: the eruption is fiercely pruritic and classically worsens at $night$, reflecting the activity of the mite $Sarcoptes$ $scabiei$ within its burrows.
Epidemiology: scabies spreads by close skin contact, so the fact that other household members have the same itchy rash strongly supports the diagnosis.
Ruling out alternatives - atopic dermatitis in this age group sits on the cheeks and extensor limbs and spares the umbilicus; psoriasis presents as sharply marginated silvery plaques rather than itchy acral vesicles; seborrhoeic dermatitis is a non-itchy greasy scaling of the scalp and flexures. None reproduce the palm-sole-umbilicus pattern with contagion.
\[\boxed{\text{Diagnosis} = \text{Scabies}}\]