All Verizon 4G LTE phones have SIM cards. That is how they authenticate to the netwok.
I take the SIM card out of mine because if the buyer doesn't know how to reprogram the SIM card for their account, it will show up on my account. Eventually the system kicks out SIM cards that have not be active in awhile, but yours was just activated so it will still be active and associated with your account. He can associate it with his account if he knows what he is doing, but if he doesn't, it could be a problem for you because it will put the 2GB data plan charges back on your account if he does it wrong. It can also kick your dad's phone off of the line and then it would have your dad's number ringing to the iPhone 5c you sold, using your voice minutes and charging you for data including any overages.
Now if that did happen, you just call Verizon and they can disassociate the SIM card from your account. They might even reverse any charges, too.
If the buyer doesn't live near a verizon corporate store, he can ask Verizon to mail a nano SIM card to him. Tell him to do that now if he needs to so that he will have a SIM card when he gets the phone. If he lives near a verizon corporate store, he can just go there and have them do it. Non-corporate verizon authorized dealers can charge for the SIM cards.
it doesn't come apart. There is a SIM card tray that pops out of the side. You can use a small paperclip in the SIM card tray hole to pop the SIM card out.
Just type "removing SIM card iphone 5c" into an internet search engine and you will find videos on how to do it.
iPhone and iPad: How to remove the SIM card
Be sure and put the empty SIM card tray back in the iPhone. Verizon doesn't have SIM card trays to give out, just SIM cards.
he can call Apple if he wants to and give them his name, but the iPhone warranty goes with the device, not the owner. Note that this is different from many other manufacturers whose warranties are not transferable to new owners. Like Samsung. With Samsung products the warranty is not transferrable to a new owner unless a state law mandates it.