You should be able to do that through your My Verizon using the switch devices option. Then you just put in the information (Device ID and SIM card number) and it will walk you through activation. The only thing you might need is a SIM card for the iPhone. My sister had an old Android Razr M and I got her an iPHone 5c for Christmas last year. I had to go in store and get her a new SIM card (Verizon direct stores do those for free though!)
The DROID Mini and iPhone 5 both use nano SIM cards, so you can just move the SIM card from the Mini to the iPhone. Once the SIM card is moved into the iPhone and it's powered up and connects with the network, you should be good to go. If you run into issues, you can use the on-line activation or contact Customer Service.
Although the SIM may in fact be the nano SIM and fit I'd still suggest getting a new one. Reason being is there are sometimes compilations with using an NFC SIM that VZW used in Android NFC smartphones when inserting into iPhones (although from what I've read it will still work and the issues are not widespread). My friend had a challenge with voicemail after putting the NFC SIM into her iPhone. You can request a new SIM via My Verizon or pick one up in a VZW store location.