Because she passed some, it's possible she could pass the rest, with help, but I'm most concerned about the rubber backing. You said you saw some of the rug fibers in her stool, and that's easier to pass because it's soft. You can try some cat hairball remedy, like Laxatone or Petromalt, or just plain Vaseline, if you have it in the house. Also, add a little mineral oil or vegetable oil (1/2 tsp.) to what she's drinking, if she's not eating and/or because when she eats, you said she's vomiting. You can also put 1/2 tsp. of Metamucil in some canned food, but again, if she's vomiting, she may not keep it down. Plain canned pumpkin (not pie filling) is good, to get some natural fiber into her, but again, it needs to be mixed into a small amount of food, and also, it's not a quick fix for constipation, as it takes a week or two to see the beneficial effects and now, you need her to get rid of the carpet material, immediately.
Sometimes, none of the traditional home remedies to alleviate constipation will help, if there is an obstruction from ingestion of a non-food item, as you describe, so it would be best if you could get help in person, from the vet. Continue feeding her liquids, as this can only help, as long as she's not throwing them up. If she continues to vomit, nothing can go down, because she can't get rid of her obstruction.