I have purchased many iPhones, but never done it online. Confused how it works.

Ziatron wrote:

If I purchase an iPhone 12 online, how does the carrier (Verizon) put my old phone number into the new iPhone 12?

Yes. Basically, when you order online, you will be given the choice to select your cellular provider, as well as, retaining your current phone number. When the phone arrives, it will already be activated and you just need to run through the new setup process. That process will provide you with the option to copy your old phone's data to the new phone.

There's 2 ways you can do this.

1. Select Verizon as your carrier during purchase, and follow the prompts to have your service setup on the new iPhone.
2. 
3. Transfer your sim card from the old iPhone to the new one when it arrives.

FWIW. I have always ordered & upgraded my iPhones from the Apple online store. In my case I use AT&T as my cellular provider, but it should be the same with Verizon. In all cases the new phone arrives activated with the same phone number as the phone comes with a physical SIM card. The old phone still works during this transition period, but I just use it long enough to both transfer & verify the transferred data to the new phone. Then remove the old phone's SIM card, and finally, restore it to factory settings so I can use Apple's Trade-in process to send it back to Apple.

As mentioned in the Image above: "Your current phone will work until you setup your new iPhone".

Once you go through the setup, your new iPhone will be activated and the service from the old one stopped at that point, but not before.

You remove the SIM from your old iPhone and put it in the new iPhone. If you want to use the eSIM you call Verizon.