When my children have been sick, especially with a stomach sickness. We usually do nothing until they stop vomiting. Small ice chunks to suck on if they can handle it.
Then we get a bottle of Vernors–has to be this kind. It is ginger ale. I let the carbonation out of it, by opening the bottle and letting it sit. I give them small amounts of this to sip on.
Then we go to saltine crackers or homemade bread toasted with a tad of butter. I like to cut them up into small chunks to make eating easier.
Our next step is applesauce.
I remember the doctors calling it a BRAT diet. Bread, rice, applesauce, and toast. If I have bananas I would try and give that as well.
Simple plain foods until their stomachs can adjust to regular foods.