When you transfer your old SIM card in your new iPhone then it will transfer the Verizon service into your new SIM Free iPhone. The service will then automatically transfer and you will not have to contact Verizon.
Note that changing the SIM card out will not transfer anything else from your old iPhone. You will need to transfer your data to the new iPhone by using one of the two methods listed below-
Transfer Data from old iPhone to new-
Do you have a backup created either on iCloud or on your computer?
Method 1 - Backup your old iPhone on your computer like this ➔ How to back up your iPhone, iPad, and iPod touch - Apple Support and then Restore your new iPhone from that backup like this ➔ Restore your iPhone, iPad, or iPod touch from a backup - Apple Support
Method 2 - The Quick Start method and here is how to do that ➔ Use Quick Start to transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support
Axel F.