Yes, there are wizarding illnesses. Two of Harry's grandparents died to one of those. This information comes from a 2005 interview with J. K. Rowling that Slytherincess mentioned in an answer.
What about Harry's family — his grandparents — were they killed?
No. This takes us into more mundane territory. As a writer, it was more interesting, plot-wise, if Harry was completely alone. So I rather ruthlessly disposed of his entire family apart from Aunt Petunia. I mean, James and Lily are massively important to the plot, of course, but the grandparents? No. And, because I do like my backstory: Petunia and Lily's parents, normal Muggle death. James's parents were elderly, were getting on a little when he was born, which explains the only child, very pampered, had-him-late-in-life-so-he's-an-extra-treasure, as often happens, I think. They were old in wizarding terms, and they died. They succumbed to a wizarding illness. That's as far as it goes. There's nothing serious or sinister about those deaths. I just needed them out of the way so I killed them.
Rowling expanded the back story to tell that the wizarding illness was dragon pox. You find this from a writeup about Harry Potter's ancestors on the second Pottermore published in 2016.
Fleamont and Euphemia lived long enough to see James marry a Muggle-born girl called Lily Evans, but not to meet their grandson, Harry. Dragon pox carried them off within days of each other, due to their advanced age, […]
ThruGog also mentions spattergroit, a scary magical illness. We find out some details of that from Harry Potter and the Deathly Hallows chapter 6. Ron needs distraction to delay the Ministry from finding out that he's gone on a quest with Harry Potter. For this, he transforms a ghoul and his family will lie that the ghoul is Ron infected with spattergroit.
A horrible half-sucking, half-moaning sound came out of the square hole, along with an unpleasant smell like open drains.
‘That's your ghoul, isn't it?’ asked Harry, who had never actually met the creature that sometimes disrupted the nightly silence.
‘Yeah, it is,’ said Ron, climbing the ladder, ‘Come and have a look at him.’
Harry followed Ron up the few short steps into the tiny attic space. His head and shoulders were in the room before he caught sight of the creature curled up a few feet from him, fast asleep in the gloom with its large mouth wide open.
‘But it … it looks … Do ghouls normally wear pyjamas?’
‘No,’ said Ron. ‘Nor have they usually got red hair or that number of pustules.’
Harry contemplated the thing, slightly revolted. It was human in shape and size, and was wearing what, now Harry's eyes became used to the darkness, was clearly an old pair of Ron's pyjamas. He was also sure that ghouls were generally rather slimy and bald, rather than distinctly hairy and covered in angry purple blisters.
‘Once we've left, the ghoul's going to come and live down here in my room,’ said Ron. ‘I think he's really looking forward to it – well, it's hard to tell, because all he can do is moan and drool – but he nods a lot when you mention it. Anyway, he's going to be me with spattergroit. Good, eh?’
Harry merely looked his confusion.
‘It is!’ said Ron, clearly frustrated that Harry had not grasped the brilliance of the plan. ‘Look, when we three don't turn up at Hogwarts again, everyone's going to thin Hermione and I must be with you, right? Which means the Death Eaters will go straight for our families to see if they've got information on where you are.’
‘We can't hide my whole family, it'll look too fishy and they can't all leave their jobs.’ said Ron. ‘So we're going to put out the story that I'm seriously ill with spattergroit, which is why I can't go back to school. If anyone comes calling to investigate, Mum or Dad can show them the ghoul in my bed, covered in pustules. Spattergroit's really contagious, so they're not going to want to go near him. It won't matter that he can't say anything, either, because apparently you can't once the fungus has spread to your uvula.’
‘And your mum and dad are in on this plan?’ asked Harry.
‘Dad is. He helped Fred and George transform the ghoul. […]’
The book doesn't explicitly say that this illness is exclusive to Wizards though, so it's possible that it's just another name for an illness we know.