So I called them and sat on hold until someone finally picked up and told me that my order was flagged for some reason. After I confirmed my address (?), they seemed to confirm my order and then hung up. I still haven't gotten any confirmation that my item shipped yet.
In general I try to avoid ordering from Amazon, but if I had done that instead, I would have my item in my hands right now. Instead I had to waste my time playing phone tag, and I'm still wondering whether or when they'll actually ship my item. All because the vendor is apparently super-averse to credit card fraud and would rather push me to Amazon for future purchases than take whatever chance they'd be taking by sending me what they've already charged me for.
I'll have to be calling them again after the weekend to see if they've actually sent it. If they haven't, I'll tell them to cancel my order. And then I'll have to wait for them to refund the charges. If they don't, then I'll have to make another call to my bank to reverse the charges.
All the while, I still don't have my item. So if you don't pay for fraud in lost cash, you're definitely paying the overhead of vendors trying to avoid being "stuck holding the bag" by throwing arbitrary verification processes in your face, introducing delays and possibly leading to you needing to make multiple phone calls to resolve problems while being left without the thing you need.