Thank you for your question.
It depends on the type of spider, and if the spider bites her before she kills it. Most of the time, cats kill and eat spiders without suffering any ill effects.
If it was a poisonous spider, and it bit her, she would have a swelling at the spot of the bite.
There are dozens of reasons for a cat to vomit, and have diarrhea. It may or may not have anything to do with the spider.