I know it's impossible to believe, but I'm going to put this out there anyway. I have talked with Verizon WIreless Customer Service on Facebook, and they said the "iPhone Calls on Other Devices" is compatible with both postpaid and prepaid customers. The error that says, "Sorry, we ran into a problem with your order. Please call Customer Service...from a different phone to complete your order. Close this page to continue," shows up because the account in quesiton is not provisioned for Wi-Fi Calling services. That can be fixed by calling the appropriate Customer Service line, or *611, if you don't know the appropriate number. Here is a word of advice to everyone (including myself): always take the advice of Verizon Wireless Customer Repesentatives and those people who work in a Verizon Wireless Store or a Verizon Wireless Authorized Independent Retailer. They know the ins and outs of how Verizon Wireless service works.
Also, the Calls On Other Devices feature (which allows calls to be made and receeved on a Mac, iPad, or iPod touch, even with the iPhone turned off) would not even appear at all on iPhones connected to Verizon Wireless through its prepaid service plans, if it were not actually supported.
Edit: If you want to set up your iPad, Mac, or iPod touch to make and receive calls without the need for your iPhone to be turned on and nearby, your account needs to be provisioned for Integrated Calling. When you mention Wi-Fi Calling to Verizon staff, they will assume you are talking abut Wi-Fi Calling for your phone.