Vermex is not effective against worms, it’s more of a tonic for those who want to spend their money on it. Personally I never use it. Healthy birds kept in clean conditions with suitable food never seem to need tonics.
The only effective treatment licensed for use in poultry that doesn’t require egg withdrawal is Flubenvet. You can buy pre-treated pellets, which are good if you have a lot of hens, but with a very small flock it’s best to get it in powder form and just mix it into their feed, which is easy to do in small amounts. If you buy a bag of treated pellets they most probably won’t finish the bag and it will go out of date before it’s time for another treatment, so will be wasted. The powder keeps until needed - a pot contains enough to treat 20 birds, so will last me a couple of years for my 4-5 girls. There are several posts about how to mix it on here.
Alternatively, you can get a worm count done by sending away droppings from each hen, and the results should tell you whether worming is actually necessary. This is recommended by many people because you reduce the risk of the birds becoming immune to the medicine if it’s given too frequently when not strictly necessary. It’s up to you. If you find out they have worms you then have to treat them anyway.
If your birds are kept in clean conditions and don’t have too much access to earthworms from free ranging, then worms won’t be a big problem, especially if you’ve never had hens in the garden before. I do always treat new pullets once they’re eating well enough to ingest a full dose, because they will have come from a breeder where there have been many generations of birds in close conditions, possibly for years, and so there’s a bigger chance they will arrive with their own quota of worms from a different biological environment.
All chickens do get worms and healthy birds will cope with a reasonable level without showing any adverse signs. But if you let it get out of hand, the worms multiply in their gut and take nourishment from the bird, resulting in low egg production and other health problems. I do mine twice a year, in September when the egg season is slowing down, especially on older birds who are in moult, and again in March to help them face the breeding/egg laying season.
If you look in the Pests and Predators section there is a lot about worms and how to feed Flubenvet. Eg this thread - you’ll see I changed my mind about how to do it after a few years!
http://poultrykeeperforum.com/viewtopic.php?t=8714
And this thread gives different points of view; http://poultrykeeperforum.com/viewtopic.php?t=10075
Also see the Health section on the main Poultrykeeper site for more horrific details and pics. (You’re having a bit of a day of it with nasty pics, aren’t you? Sorry about that!)
https://poultrykeeper.com/?s=Worming+chickens
https://poultrykeeper.com/digestive-system-problems/