The big problem with the iPhone 4 is that it is a 3G device. All of the major wireless carriers are going to be shutting off their 3G networks, and once that happens, the phone will become a Wi-Fi-only device.
I assume you actually have a 4S, since the 4 is a GSM-only device, and is incompatible with Verizon’s 3G (CDMA) network.
You can’t just move a SIM card. Verizon’s 3G network is CDMA, and doesn’t use SIM cards. So you’ll need to get Verizon to send you a SIM card for your new phone (or you can buy your phone from Verizon, and they’ll ship it with one pre-installed).
You can keep your existing plan if you want to. You can transfer your number to the new phone when you order your phone (or a SIM card). The old phone will be deactivated when your new phone is activated (as a part of the activation process after you get the new phone).
If you are willing to get a new plan (which may well cost more per month), Verizon may have some attractive promotions to encourage you to do so. For instance, when i upgraded my iPhone 6 to a 13, Verizon offered me $350 trade-in on my old phone in conjunction with a new 5G unlimited plan (it would’ve been worth only $10 otherwise).
The simplest approach is probably to order your new phone via Verizon’s web site. But pay close attention to everything you’re asked to click on - you don’t want to subscribe to things like new plans and insurance policies if you don’t want to. This will make it easiest to transfer your existing account (and phone number) to the new device.
The iPhone 4/4S are 3G devices. The iPhone 5 introduced 4G/LTE for data-only. The iPhone 6 introduced VoLTE (voice-over-LTE) and is therefore the minimum iPhone that will be compatible with wireless networks once 3G services are turned off.
Unfortunately, moving the SIM isn’t really an option.
Verizon’s CDMA network doesn’t use SIM cards (unlike their LTE network, which does). I think Verizon iPhone 4S devices included a SIM card (I have an old card from mine), but I think it is only used when roaming to a GSM network. It may not have the data needed to run a modern phone on an LTE network.
Furthermore, an iPhone 4 uses a micro-SIM, while more recent iPhones use a nano-SIM. So you’d need to cut the card in order to make it fit.
But there’s no problem telling Verizon to activate a new phone on your account with your existing number. They’ll issue you a SIM card (or if you buy a phone from them, they’ll pre-install it in the new phone for you). The new SIM will take over your account when you activate it in a phone.
Yes, Verizon will charge a $35 “activation fee” in order to do this. I don’t think they will impose the charge if you just move your SIM card, but they warn you not to do this, because new phones support new features and your old SIM won’t have the data required to use them. At least that’s what they say - I’ve never tried just moving a SIM, so I don’t know how accurate that is.
But in this case, upgrading over 11 iPhone generations (5, 5S, 6, 6S, 7, 8, X, XS, 11,12, 13) and moving from 3G to 5G networking, I would expect a new SIM to be required.