It is advisable to de-worm your cat under veterinary supervision. The reasons are (1) no medication is effective against all types of worm and therefore a proper diagnosis is required and (2) medication needs to be given precisely as directed by your veterinarian and (3) “natural” side effects such as vomiting and diarrhoea need to be distinguished from toxic reactions and (4) deworming pills are poisonous but more poisonous to worms than cats! However, as mentioned, the medication needs to be administered precisely.
Most cats, from time to time, will have worms of some sort. If your cat is an outdoor/indoor cat it is likely that he will have worms especially if he is a hunter because he’ll acquire worms from prey. However, even indoor cats can get worms from fleas which transmit the tapeworm to cats.
I am sure that the majority of stray cats wandering around the environment have worms and they survive quite nicely because the ideal parasite lives on its host without causing serious ill-health for obvious reasons.
However, worms can cause problems if their numbers are sufficient. Tapeworms, hookworms and roundworms are the most common intestinal parasites and cats.
My cat used to be an outdoor freeroaming cat and he acquired tapeworms. You can tell when a cat has tapeworms because they have what looks like a grain of rice attached to their bottom. This is a segment of a tapeworm and it will be moving.
Once your veterinarian has diagnosed a worm infestation and prescribed pills you can act alone to administer the pills according to his or her directions. You can request a repeat prescription from reception without attending the veterinary clinic and without taking your cat to the clinic. This is my experience in any case.
I feel quite confident in saying that a lot of experienced cat owners in normally multi-cat households probably diagnose worms and administer medication without seeing their veterinarian. This is because worms are quite commonplace and therefore by default cats are given deworming pills as a precaution.
As the pills are effective against more than one species of worm (but not all worms) they act as a preventative medicine. Experienced cat owners i.e. cat rescuers keep cat medicines on hand and know feline medicine and health quite well. Therefore the statement in the opening paragraph is not altogether true in practice. That does not mean people should avoid their vet.