Sounds like a browser problem.
First off, which web browser are you using? Typically the Disney site recommends either Firefox or Internet Explorer (MSIE).
If one of those, it's most likely a cookie setting.
In MSIE (note, these steps are for MSIE 7.0, if using another version they should be similar, but may not be exact):
Go to Tools -> Internet Options
On the General Tab there's an area called "Browsing History" with a delete and settings tab, click Delete
On here there will be several options, delete the Temporary Internet Files as well as Cookies.
Click OK then reload the page and relog in
In Firefox (note: this is using 3.5.3, like above, steps may vary, but they should be similar)
Go to Tools -> Options
Click onto the Privacy tab
Click the "Show Cookies" button
Click the Remove All Cookies button
Click OK to close out the windows
Relog into the Disney page and try again
If this still doesn't work, it's likely settings somewhere on either of those pages relating to how cookies work. Make sure to have them set to allow cookies as well as allow third-party cookies (if available). You can change this setting back to what you had it after if you so desire.
If none of that works, then it could be either a setting from your network (could be typical if at work trying to do it, far,
far less likely from home if you didn't set that up.) I don't know enough about networking to have a solution to that though, and chances are, if at work, the IT department did it for a reason and would be unlikely to adjust it for this.
ETA:
I forgot the original reason for replying ><, I did notice that after I booked Yak & Yeti today (via online), I went through and added the other 11 confirmations I got yesterday (via phone) and it only was sending me Yak & Yeti. I haven't tried relogging in and out yet, to try again, but we'll see. At least I have copies of my ADR confirmation numbers both by hand (when I was talking to her) as well as in my planning doc (which is housed on my work comp, my home comp, as well as my inbox from when I sent it over to home

).