I don't think you should be expecting a USB webcam (certified to work with Windows and Mac ) to be able to work on any Android device regardless of custom ROM. Windows computer have generic drivers built in to let the operating system handle external imaging devices. Android does not and doesn't have any framework for you to use external imaging devices.
Might be worth loaning a laptop from a friend or getting a used one if you normally use desktops for everything.
If you want a really cheap laptop with a decent build Hp has their stream series for like $250 from the Microsoft store.