Thanks for your query.
It seems the boy is suffering from tummy bug ,the viral infection of stomach or gut.
In view of the fact that his vomitings are not freq , the stool consistency has improved and he is in good form , make any serious pathology very unlikely.
Still if his stools are persistently very pale ,white or clay coloured for next 2-3 days, please see your GP as it might happen in cases of lack of bile (a green fluid that digests fats) in stool secondary to any liver
Wishing him speedy recovery.