According to a Verizon CS rep I spoke to, it should be possible to use the "Upgrade Device" procedure to upgrade my current device to a Moto X 2nd Gen, and, instead of the black or white 16Gb models available through Verizon, choose "Custom". After checkout, Moto Maker should send me a PIN which will allow me to design my device on Moto Maker, and the device will be associated automatically with my Verizon account.
"Custom" is currently not an option, and previous forum posts indicate that this is not a new problem.
I hear you asking, "Why not just buy it on the Motorola site?" I'm a path-of-least-resistance kind of person about things like this, so I gave this a shot. Motorola wants me to give them my Social Security Number in order to provide Verizon with information with which they'll do a credit check, even though I've already told Motorola that I'm a current Verizon customer and Verizon already knows my credit history. I'm not giving Motorola my SSN; they don't need it.
If this is some sort of conspiracy to drive foot traffic to brick-and-mortars or drive CS calls, then, by all means, maintain the status quo.