Can I use an external USB Adapter with Airmon on Android? I have an Android phone and it doesn't support monitor mode internally. I have an external Alfa AWUS036H and I ordered a cable for it to plug into my phone. My question is, what app should I use to put the external adaptor into monitor mode? I am rooted, but I refuse to install a ROM or a kernel. Thanks!
Short answer is maybe.  You could grab an adapter (they are roughly $5) that would allow you to plug in to the charge port (almost always micro USB) to a standard USB port (AKA USB A):

Like this page of options

Then you could try plugging it in.  Android has been adding Ethernet as well as Mass Storage drivers to try and accommodate devices over the USB bus.

I did a little searching and it appears your adapter has been used by other Android users (probably for exactly the same air capture purpose good 'ol aircrack and/or wireshark).

Here's a video even, enjoy

VIDEO NOTE: Newer devices will most likely not need the extra power cable this guy talks about newer devices are putting more power over the port just for things like USB hard drives, and devices.