If you are using Android Lollipop or above, install Termux, install grep (if not already installed) by typing apt install grep, give Storage permission to Termux from Settings app so that Internal Storage can be accessed. And that's it. Enjoy your grep. You can also install busybox the same way in Termux and use its utilities. Installation and usage doesn't require elevated privileges.

If you are using Android version before Lollipop, Robert's answer is the way to go if you must need busybox. I just want to add that instead of doing chmod later (which you cannot because Android didn't have chmod back than with toolbox), you set the permission correctly on the binary before pushing it in /data/local/tmp/.

While I have not tried them, some grep based apps are available in Play Store and can be found here.

If you are using Android Lollipop or above, install Termux, install grep (if not already installed) by typing apt install grep, give Storage permission to Termux from Settings app so that Internal Storage can be accessed. And that's it. Enjoy your grep. You can also install busybox the same way in Termux (it appears to come installed with Termux per user iBug) and use its utilities. Installation and usage doesn't require elevated privileges.

If you are using Android version before Lollipop, Robert's answer suggests a good way but some steps can be eliminated. Simply fetch the Android binary compatible with your processor from here, rename it to busybox and push it into Android using adb as:

adb push busybox /data/local/tmp/   # don't push it into /sdcard/ as it is both unnecessary and removes executable bit from the binary which you cannot add without having busybox's or toybox'x chmod utility. 

You can now use busybox. You would want to setup an alias or change ENVIRONMENT variable of your shell accordingly to map that file.

Alternatiely, while I have not tried them, some grep based apps are available in Play Store and can be found here.