The majority of wild mushrooms are harmless. But, some are poisonous, a very few are deadly in small amounts. Unfortunately, unless you know what kind of mushroom your dog ate, there is no way for the vet to know whether it was poisonous or not. If one suspects mushroom poisoning, it won't do any additional damage to induce vomiting/pump the stomach, so often vets will do that as a precaution, even though it may not be necessary. The fact that your dog vomitted on her own is probably a good thing. If it has been several hours since she ate the mushroom, then further voiding of stomach contents probably won't do much good, since whatever she may have consumed is now in the intestinal tract. PERHAPS a laxative to encourage rapid passage of any potentially poisonous mushrooms might maybe (am I equivocating enough yet?) be slightly beneficial.
The most common type of mushroom toxin in North America is muscarine (NOT to be confused with muscimol, which is an entirely different type of mushroom toxin). Muscarine is found in several different types of mushrooms, and there is no one single feature that will tell you whether a mushroom contains this toxin or not. One has to be able to identify the species of mushroom to know. Muscarine can cause vomiting and diarrhea, and it's most dramatic symptoms are that it stimulates the sweat glands, tear ducts, and salivary glands. In dogs, this translates into lots of drooling. Lots and lots and lots of drooling. I speak from personal experience with one of my dogs on this one. LOTS of drooling. The biggest danger of muscarine is that it also depresses the heart rate. Usually people and dogs that are poisoned by muscarine recover with no long term damage. A few hours of unpleasantness, lethargy due to the lower heart rate, and then full recovery. Obviously though, anything that depresses the heart rate can be dangerous if the dosage is excessive, or the patient already has heart problems. So, it's important to monitor that. On the other hand, and this is important, be EXTREMELY CAUTIOUS about administering any sort of heart stimulant like atropine, unless it is a true emergency. Other mushrooms contain different toxins that can act synergistically with atropine and overstimulate the heart. Unless one knows for sure that the mushroom consumed contains muscarine, it may do far more harm than good to administer stimulants.