Cats are specialized for eating meat, more so than many other carnivores such as dogs. Chewing food is more a necessity when eating plant matter. Cats, being such specialized predators, even have fewer premolars and molars than dogs, because they don't need to chew so much. So swallowing pellets whole is pretty typical for cats, and not necessarily the reason for the vomiting.
Vomiting after eating is also relatively common in cats, and can be caused by a variety of things. Some common causes are, the particular food simply doesn't agree with them, or that the animal is wolfing its food too fast. It's possible that even the lack of chewing is an indirect cause, in that actually chewing the food slows your cat down. The latter can be solved by either doling out the food in smaller portions more frequently, or figuring out some way to prevent the animal from eating at maximum speed, such as putting the food in a food puzzle toy, or getting a dish designed to make it a little harder to get its food.
Examples of what's available by Amazon search: https://www.amazon.com/slp/puzzle-feeder/283z7xrmcy386pz
The other option is to simply not give your cat the treats, since they are not a necessity.