It’s a known thing that root functionality was broken for latest update for Verizon Samsung S4 – build VRUAME7 However a working root exploited for build VRUAME7 is released today by open1your1eyes0, senior member at XDA. You need to keep in mind that this exploit will not work on any build other than VRUAME7.
This seems to be a repackaged and simplified exploit based on original root exploit released by DoomLord, recognized developer at XDA. Below is the description mentioned in the original post.
This is a simple guide that will allow you to easily root your Verizon Galaxy S4 device without having to wipe your device or flash a full prerooted factory image. This method works using a root exploit (originally created by user DooMLoRD) and modifications of this exploit to work on the Verizon Galaxy S4 device (provided by user PWM978). Only change is I have modified the root method to use the latest SuperSU binaries and repackaged with simpler instructions.
You need a microSD card formatted as exFAT to perform this exploit. Head over to this thread at XDA for detailed instructions on preparing your PC & micro SD card and then root your device along with download link of actual root exploit.