Here's my story.
I took a screenshot of the "We have an iPhone for you" page with my reservation number at 3:20 AM EDT. I got the "Your iPhone has been reserved" e-mail at 6:01 AM, and "Important information about your iPhone reservation" at 6:04 AM. I was getting "the page you're looking for cannot be found" error until sometime between 12 PM and 1 PM. I was then experiencing the same thing as some other folks -- I couldn't select "Delivery" because it said Sprint might want a deposit, and I couldn't select "Pick up" because there are no local stores with the phone in stock.
I connected to a chat agent who was frighteningly unhelpful and did not understand the reservation e-mail I was asking about, e.g.:
"Okay you would not be able to have it delivered since that is a reservation for in store, to have it delivered you may place a new order but in this case you should check back often because the availability will change rapidly."
After a few rounds of the same unhelpful answers, I asked him if I would need to go to the back of the line and start over if the phone was still "unavailable for pickup" in all local stores through the expiration of my reservation (September 10 at 3 PM EDT). He replied,
"That would be the next thing to do but you will get a reservation as long as you check back often and hit refresh."
I called 1-800-MY-APPLE. I had to lie to the automated bot -- if I requested iPhone sales help, I was disconnected after being apologized to. I requested help with iPad sales and I got through to someone after about 12 minutes on hold. When I explained the situation to the rep, she said that I would need to pay Sprint the deposit that I owe them before I could proceed. I could instantly sense she had no idea what she was saying, so I cut my losses and hung up.
I started a new order for the phone (iPhone 7 Plus Rose Gold 128GB Sprint) and was able to complete the checkout process, including home delivery. No mention of any deposit. Unfortunately the delivery date is October 4 through October 10.
So the plan is to keep checking the reservation link to see if it will eventually allow me to claim the iPhone that I rightfully reserved for September 16 delivery at 3:20 AM EDT today; failing that, I will hope that the delivery window moves up on the phone, and will keep an eye out for this year's variant of the iStockNow live availability tracker map. I guess it will be a little fun to go on the hunt, and I'm hoping Sprint's unpopularity means I'll be able to find this model somewhere.
What a bummer. I almost feel bad for the customer support reps who are delivering false information to callers. I wish they could just admit they don't know what's going on and can't help.
Good luck to anyone else in the same boat as me. If you have any success with the reservation before it expires tomorrow, please post an update. I'll be checking in every so often to see if anyone gets lucky.
ADDENDUM: I was worried I would completely lose the reservation, even though it appears to be worthless, once I finalized a new order using the same Sprint phone number. But I am still able to use the reservation link and proceed to the "My Bag" page, even with the pending order. I am still seeing the same options as before, though -- no "Delivery" option because of a deposit, and no availability for pickup.