Hi there,
I also rescued my yorkie and she, too, had vomiting and diarrhea when I got her. The vet tried every premium food on the market and she just kept vomiting and having diarrhea. I did try the Royal Canin one, but it did not work for her, surprisingly. It might work for your Yorkie though.
I decided to cook for her and one of the members here recommended Ask Ariel - a nutritionist. I did this and the nutritionalist, Susan Blake Davis, told me to give my Yorkie chicken, beans, carrots and rice. This worked extremely well. All the vomiting stopped and so did the diarrhea. However, chicken is something some Yorkies are sensitive to, so it may still be trial and error for you. Susan will be able to help you with this, as she has helped a lot of people.
I have since changed vets and the new vet told me that dogs with digestive issues often do very well on home cooked food and not to try any other dog food, as she is doing well. She still has the occasional upset tummy due, probably, to her eating things she shouldn't on her walks! She can now eat fish, lamb and beef without any problem!
I know that 'Ask Ariel' has some natural treatments for IBS and, if my yorkie still had lots of problems with IBS, I would definitely try to get someone to send me these remedies (I live in the UK).
One thing I thought I would mention is that my veterinery dietitian in the UK told me not to give her any probiotics, as her tummy would probably react negatively to them. I didn't really believe him, I'm afraid, so I tried a good probiotic and (you've guessed it!) she started having diarrhea! My Yorkie is very sensitive, so you may not have the same problems.
I hope you find a solution soon. I know what you are going through and it is so upsetting to see your little Yorkie suffering. If I can be of any further help in any way, you only have to ask.
Hugs
