If you cat is strictly an indoor cat then yes, they have hairballs and you can go to any pet store and get a tube of malt-like anti hairball remedy that the cat will either lick off your finger (or if it's a fussy cat) spread it near the mouth or on top of the paws. Cats are clean creatures and your cat will lick it's paw to get the sticky substance off. There is also some good brands of food out there that stave's off hairballs It is wise to take your cat to the vets to be sure there is no other reason for your cat to be vomiting.
Generally a cat will not vomit up hairballs for days on end and some cats can have a twisted gut (bowel. ) Please take your cat into a vet to get a proper diagnosis.
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