If this was my therapist the answer is yes.

Since covid hit the world times have changed. Most therapist have to abide by state and federal health department standards around bodily fluids including throw up. Regardless of reason. In health care settings throw up is considered a toxic bodily fluid requiring special handling and disposal. You know those red labeled containers, bags and so forth that you see in hospitals doctors offices and so on.

Maybe you can contact your primary physician for a doctor's statement and carry your own toxic bodily fluids containers so that only you have to handle the throw up bags and containers, take them home with you and disposal. That way the smell and possible contamination by you throwing up in her trash can won't happen.