Have you tried diconnecting the shift cable at the trans. Can you manualy shift it into REV at the transmission? Just trying to rule out the cable.
If it shifts into REV, but the transmission doesn't move it, it could be a broken reverse chain (#3234264, $70). Usually when these lets go, it will make a loud pop. Usually happens when its in reverse (of course). Transmission has to come out in order to fix it. Hopefully, it didn't do more damage inside. Don't drive it until its fixed, cause if the chain is rolling around inside and it gets into the gears, it can grenade the trans.