How to do GREP text searching without root?
You are mixing up two things:


Installing a binary like busybox system wide from within an app (which includes grep)
Running a binary 


The default app based installers for busybox require root access to install busybox system-wide. This is true, however this is just because it is an app. Once the binary is installed you don't need root access.

Additionally even without root you can also install a busybox (or whatever) binary on your device:


You need to activate Android Debug Bridge (adb)
Copy the binary onto your phone (if it is not already there): adb push busybox /sdcard/
Copy the binary from the sdcard section (where it can't be executed) to different directory like /data/local or /data/local/tmp: adb shell "cat /sdcard/busybox &gt; /data/local/busybox".
Then make it executable using adb shell chmod 0555 /data/local/busybox


Now you can execute it using its full path from within a terminal app or inside adb shell:

/data/local/busybox grep ....