It's not yet available over-the-air, but an update to Android 4.4.2 KitKat can be had for the Samsung Galaxy Note 3 on Verizon if you're willing to plug in. The update comes courtesy of either Samsung's sync and update software Kies or the Verizon Software Assistant.
As noticed by Android Central Forum member spunkmeijer, the update took about half an hour to install.
Kies is available for both Mac and PC, as is the Verizon Software Assistant, and both are triggered merely by plugging your phone into the computer while the app is running. From there you'll be walked right through everything you need to do.
Verizon's Galaxy Note 3 is running behind on the KitKat update train, with the other major US and international carriers getting the update as of March. We don't have work yet for when the update will be available over-the-air for those that don't want to install yet another app on their computers, but hopefully it won't be too much longer.