Yes certain diseases can be spread through Vomit so if someone is sick and vomiting they should stay home from work or school until they're no longer throwing up.
And if a child is throwing up at school they should be allowed to come home until they're not sick anymore.
Vomit can spread many diseases except for HIV. For HIV to spread to someone through Vomit, the vomit would have to be bloody.
You should immediately wash your hands after cleaning up vomit to prevent the spread of bacteria and diseases.