That would make me very uncomfortable as well!
I am going to be completely honest, I don't know that I would trust any reassurance I got here on a message board. For an issue of security such as this, I would only really trust what I saw (https, etc) or information obtained directly from Disney's IT department. (I'd also keep a good record of who gave me that assurance).
Actually, if it doesn't look secure (https), I am not sure that I'd even trust it if Disney assured me. I am quite paranoid about stuff like that.
Thanks for the warning though, it's a good reminder to make sure you look at things like this before giving your CC number online.