This has happened way too many times the last few months. The first time was when I was at Disney trying to access it for an immediate use. Disney IT...
Do you have a cell phone number associated with your Disney account? When I sign into the gift card site, it gives me an option for a text to my cell number or an email. If not, go to the WDW website, at the bottom of the page click show more links. Under My Disney Experience, click My Account. You can edit your information & add a cell number. Then try to sign into the gift card site & hopefully it will give you an option for a text.