Is there a way to log into and root my device without using ADB?
There are several ways to achieve root access on an Android device -- and they differ a lot, depending on manufacturer, device, Android version, and even Linux kernel version running. Unfortunately, devices don't ship with "root" accounts (as we are used to from our desktops and laptops) -- but there's no common way to "root" them.

Rooting is mostly done via Exploits. Some require ADB installed and an USB-Session established. Others do not require ADB. And again others ship as a single .apk to be copied to the device and started there (so-called "one click root"). And again another method might involve a custom recovery being flashed...

So many methods. But to explain them all in detail, this is not the right place -- while your question should be answered already, as it required only a "yes" or "no" :)