I have a 9 week old puppy has been wormed twice since I got her when she was 4 weeks old. She still has short flat worms in her poop and this morning I noticed she wasn't acting like herself and she was throwing up. Can this be caused from the worms?

Small flat worms sounds like tapeworms and are pretty harmless. More gross than anything else, nevertheless do need treating. But a vomiting puppy should be seen by a vet. The vomiting can be caused by any number of things, from parvo to having something lodged in the digestive tract. Better safe than sorry in the majority of cases. A young puppy can become dehydrated quickly and go downhill pretty fast.
My suggestion-get her in for a check up.