Puppies from all dog breeds will occasionally throw up, as will adult dogs from time to time, but from our experiences, it seems that Frenchies throw up more than most other breeds. The fact that they are brachycephalic (short faced) probably has a lot to do with this. Frenchies will tend to gulp water and their food (which is why they are sometimes [OK, often times] gassy), and this also contributes to their propensity to throw up more than most other breeds.
Based on our experience, if your puppy is throwing up a time or three a day, especially early on as they are getting accustomed to their new digs, 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, and feeding your puppy smaller, more frequent meals can help. 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. We had one adult female which had a problem digesting the dry food, which we feed to all of our dogs, which has rice as the only grain, in addition to all of the other ingredients. We put her on a grain free diet and she did very well, eliminating the problems of throwing up and having loose stools. We also had one young Frenchie which had a problem with all of the dry food we tried, including grain free, so we tried her on a raw diet and this did the trick.
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 that there is a problem, corrective action should be taken. It very well may be as simple as a diet change, but if this does not solve the problem, your vet should be consulted.