I totally agree with you, I'd rather clean up my own messes, but...
There are all kinds of "Barf Bags" out there, some you can get at Walmart or CVS. But the ones that I'm familiar with, that I recommend are:
https://www.amazon.com/Safetec-Vomit-Bag-4-75-3-25/dp/B0198ZL6QE/ref=pd_lpo_121_lp_img_4?ie=UTF8&refRID=ZZAFCKTM8A6539R3ZTSP