We once had a cat left on our doorstep at work, who turned out to be hyperthyroid, which has vomiting, diarrhea, and weight loss as the main signs. The boss decided to do the surgery to remove the overactive gland -- they have one on either side of the neck, so if one is overactive, you can take it out and the other one can still do the job just fine. Shaved her neck prior to surgery...and found she already had a scar from one side being taken out. We figured her owners paid for surgery the first time, but couldn't afford to get it done a second time. (If you ever have a hyperthyroid cat, getting the I131 treatment can get to overactive tissue even if it's ectopic in the chest, and leaves the surrounding normal tissue alone even within the same side.)
She did the surgery anyway, BTW, and sure enough, she had enough thyroid tissue left to be perfectly fine, was adopted by one of our kennel kids.
Do not leave a pet on anyone's doorstep, ever....but if you do, please leave a note with basic info like age and medical history!