Does the existing SIM card in your current phone fit in the new phone? Otherwise, yes, you can visit a corporate store and get a new SIM card for the phone.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
Unless your current SIM is difference size you can just switch t to the new phone. Unlocked? Like not a Verizon branded phone? Better make sure it actually works on Verizon. Which phone is it? As Ann says go to a CORPORATE store not one that says "Authorized Retailer or "Premium Retailer" they will charge you for a SIM. Corporate stores won't.