How do I restore my disabled account?

If your account has been disabled in the App Store and iTunes, it isn’t something that you can fix yourself. You will need to contact Apple Support. The following Apple article includes a link that you can use to contact Apple Support. Click the “request reactivation” link in: If a message says 'Your Media & Purchases account has been disabled' - Apple Support

This is primarily a user-to-user support forum; other Apple users like you.

There appear to have been recent changes where self-help may be possible. Since I have never had this problem I don't know what tapping "Continue" does, and if it just takes you to the contact methods or does something else.

If a message says "your account has been disabled in the App Store and iTunes Store" or "You cannot login because your account has been locked", you might be able to request reactivation by tapping Continue right from the alert. If that does not work, Apple provides a link with three options for this specific issue. Tap or click here and choose the self help method, or a contact option ➞ https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362 .

Thanks Limnos; I'll update my answer.