A call option allows a person to buy a stock at some future date for a specified price. By implication, if it is "in the money" you'll be able to buy the stock at a potentially deep discount.
So it seems like the financial stability of the option writer might be relevant, both due to the potential that they'd no longer be in business when the time comes, and to the potential that they might not be able to afford the transaction if they are.
However, reading about options I don't see much discussion of these risks, suggesting that they've been solved or mitigated somehow. Is that true, and if so how? Is it because the stock comes from the market maker which is a relatively stable company which would have shares available? That's just a guess, I don't know if that's how it works.
(One could ask a similar question about put options but I've focused on call options here.)