If you do online check-in (assuming you can--my last trip the system wasn't working and it wouldn't let me!), you will be asked to enter your CC number, but you do not have to provide it.
Nor do you need to give your CC number at the front desk if you check in in person.
Disney does like having your CC...but doesn't require it.
I've gotten in the habit of simply not ever using my CC for room/check-in. If I want to buy something, I use cash, a gift card, or my actual CC...KTTW card charging privileges can be an advantage--e.g., if you are paying for someone else in your party (e.g., my college-age son), but it's not "necessary."
The online check-in process does not make it obvious that you can proceed w/out entering your CC info...and if you skip that step, it will ask you again...but your check-in will proceed (if it's going to anyway--Disney's IT is such that as often as not it isn't working!) w/out CC info being entered.
Happy trip!