I'm sorry to hear that your pet is ill. There are many types of palm plants, and some are more toxic than others, however the typical indoor palm plant leaves, other than those known as "Lucky Bamboo," are relatively non-toxic. They do contain agents in their leaves however that can cause vomiting and diarrhea. The sago palm is another plant that you should be concerned about - ingestion of its nuts or sometimes leaves, can be extremely toxic. In some cases where the plant is non-toxic, the vomiting and diarrhea will not stop, just due to the irritation caused by the plant. This may require treatment by a veterinarian tohelp administer medications that will help stop vomiting. Unfortunately there are no good over the counter medications for vomiting in cats. A small amount of milk may help to coat the stomach however.
I hope that this is helpful, as I could not get my plant pictures to load.
The taller, more leafy palms are usually non toxic. In most cases, the vomiting will stop after a short period of time. If your pet continues to vomit, expecially multiple times, or cannot keep food or water down, then injectable anti-vomiting medications are probably necessary.