Puppies/ Adults from all dog breeds will occasionally eat or drink to fast and tend to be gassy or vomit from time to time due to this. It seems that Frenchies throw up more than most other breeds. Because they are brachycephalic (short faced) probably has a lot to do with this in short nose breeds. Frenchies will tend to gulp water and their food (which is why they are sometimes or many times gassy which also contributes to their propensity to throw up more than most other breeds.
Based on experience, if your puppy is throwing up a time or three a day, especially early on as a pup and are getting accustomed to their new living quarters it is nothing to worry about. Especially if your puppy seems happy and healthy and is not losing weight. If they do gulp food and water, there are some food and water bowls on the market which will slow them down that can help. Feeding your puppy smaller more frequent meals can help slow them down and help eliminate some of this issue. Sometimes having dry food always available to them can also help as they will not feel a need to rush and gulp their food.
As puppies age, they should throw up less and less frequently until it becomes a fairly rare occurrence. If they continue to throw up several times a day, there is a good possibility that your puppy might have a food allergy. Some may have a problem digesting dry food, which most will feed to there dogs that contain grain. Putting on a grain free diet might do the trick eliminating the problems of throwing up and having loose stools. Some may also have a problem with all of the dry food including grain free and need to be placed on a raw diet.
The bottom line is that almost all Frenchies will occasionally throw up, so it is nothing to be alarmed about. However, if it is persistent and there are other signs(weigh lost, no energy, diarrhea or dehydration, corrective action should be taken right away . It very well may be as simple as a diet change to fix vomiting, but if this does not solve the problem, your vet should be consulted.