Yes, if you purchase a Verizon phone directly from Verizon or an authorized 3rd party retailer you will be charged an upgrade fee. If you purchase the device online, it will be $20. If you purchase it in a store or by calling, it will be $40.
To avoid this fee, you can simply purchase an unlocked device directly from the manufacturer(Apple, Google, Samsung, etc...). Make sure you are getting the unlocked version which will not come with a Verizon SIM card. You can simply take the SIM card out of your old device and insert it into your new device.