If kid is sick for unknown reason it should be examined and its not overreaction.
sending them to class risks other kids getting sick or your kid throwing up on class again making things worse.
+ I would never choose the shelter from bullies option because that is sure overreaction while there is nothing to suggest your kid is being bullied or should get bullied for being sick
and while your kid might not be bullied yet they surely will be if teachers start treating your kid special and giving " do not bully the sick" lectures making a thing over it.
bullies feast on that nonsense.