Although the first mission begins at 11 am when the World Showcase opens, new recruits can join Team Possible as early as 9 am.
What time will my mission begin?
If you sign up right at 9 am, it will be 11 am. Beyond that, it's not possible to say.
Because it's like a FASTPASS system, the Kim Possible World Showcase Adventure offers return windows (which shift in 15 minute increments as opposed to 5) and once it assigns a certain number of groups, the return window gets pushed back 15 minutes. For example, the Mission Pass shown above was obtained at 9:53 am (as you can see by the time printed in the lower right corner) on as low day and it's for 11:15 am. It's also reasonable to expect as 11 am approaches and guests begin to head towards the World Showcase, the number of Mission Passes distributed will increase exponentially.