Do I need to root my phone to get a tcpdump?
Yes, tcpdump is an administrator-level application (I highly doubt there will ever be an Android permission level to view all other apps' network traffic). Once rooted, if you have tcpdump binary (compiled for your device's cpu type) on sdcard, try:

su
mount -o remount rw /system/
cd /sdcard
cp ./tcpdump /system/xbin


Now you can use tcpdump command.

Without rooting, you could intercept traffic on your wifi LAN by setting a proxy to your computer in your device settings, and installing mitmproxy in a virtualenv on your computer and running it.