You cannot do it via the website - but you CAN do it from within iTunes. There IS an option for "none" when selecting a credit card. (even after the account has been active for a long time and used)
Once logged into iTunes, within the Store, on the right sidebar area, there is a section called Quick Links. There is an option there to click on called "Account".
You will see "payment type" and a link to "edit". Click "edit" - then at the far right on the list of credit card types you can select, there is a gray icon that says "none". Select this, and save.
I hope this helps.
From reading other posts, I thought it was not possible. But, I called apple (to complain) - and they actually helped me with the above information. I'm glad they did. The same options were NOT available on the store website.