Yes the dirty litter box could make your cat sick especially if more than one cat uses the same litter box.
The poop and urine can contain bacteria that can be transferred from one cat to another when they step into the litter box and make contact through their paws from the litter and poop and urine.
Then also it can be transferred to them when they lick their paws which they do from time to time.
Also a litter box that is not kept clean may cause the cat to decide not to use the litter box and may cause the cat to hold in their urine and poop which can cause health issues for the cat.
And the smell of the poop and cat urine in the litter box can cause health issues in humans as well as it's not healthy to breathe in the poop and pee smell from the cats litter box.
So make sure you scoop the poop out of the cats litter box daily or twice daily or more often if more than one cat uses the litter box.
Change the cat litter every few weeks to at least once per month to keep the cat litter clean.
Or consider spending the money and getting a robot litter box which cleans itself after the cat uses it to keep the cat from becoming sick and keep yourself from becoming sick.
You still need to empty the cat litter and poop and put fresh cat litter in it but the cat litter box robot helps a lot at keeping the litter box clean.