I was wondering if it could have been a bee sting. The fact that there is no swelling kind of indicates it is not a bite. If a little dog did get bit by a poisonous snake I doubt it would even live long enough to make it to the vet. Garden snakes rarely bite and when they do the most you would have to deal with is a topical infection. A bee sting could cause some issues especially if your dog has an allergic reaction but I would think that there would be some swelling at the site.
Do you have rose bushes or something with sharp edges in the yard? Maybe he managed to eat a bug or plant that made him sick. It's all just speculation at this point. Did the vet give him something for the vomiting? Has the vomiting totally stopped?
We have garden snakes around here. We live down in a wooded area. I only let Gracie play in the part of the yard that is kept mowed short and I keep her out of the garden and the edge of the woods. I'm sure she would love to play with a snake.
I hope your baby is feeling better real soon.