Oh boy, new Amex web interface.
I don't see it at all. I think if you want to change it at this point, just chat with them.
@Anonymous wrote:I wonder, how far out is your due date?
I just checked mine, and this is where I found it.
Click on the Payments tab at the top. That should take you to the "Pay My Bill" section. Look on the bottom right, just below the Payment Date dropdown, and you will see a link for changing the due date.
Hrm I had gone there yesterday but didn't have a link; kicked a couple payments and now I have no ability to make a payment on 2/3 cards, and the 3rd hasn't had a balance in several months and there's no option there either.
I last changed my dates a year ago but I don't remember it's being awkward or difficult to find until this new interface for me.
@vanillabean wrote:It would appear that the card had already entered the phase of preparing for closing the monthly billing cycle last night. Once the statement was posted this afternoon, the change due date link reappeared. Well sort of, it did on a computer, but not on a tablet using the same web browser!
Oh wow... the option was available from a computer but not a tablet... thanks for that heads-up! 
Side note: I'm not a fan of the new Amex site (hopefully it will grow on me in time).