Most smartphones that Verizon supports have access to call and text history. Are you talking about when the records have been deleted off the phone though?
Basically, all handsets have a call and text history - but where it's located and how you access it will vary from phone to phone.
The best thing to do is get familiar with the phone in question, regardless of whether Verizon is your carrier.
For example, if you have a HTC One, then read this: Checking calls in the Call History
Or if you have a Moto x consult this user guide: https://motorola-global-portal.custhelp.com/ci/fattach/get/680258/1377800406/redirect/1/filename/Xfon_UG_Vzw_68017470001A.pdf
